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WELLS"    MATHEMATICAL   SERIES. 


Academic  Arithmetic. 

The  Essentials  of  Algebra. 

Academic  Algebra. 

Higher  Algebra. 

University  Algebra. 

College  Algebra. 

Plane  Geometry. 

Solid  Geometry. 

Plane  and  Solid   Geometry. 

Plane  and  Solid  Geometry.     Revised. 

New  Plane  and  Spherical  Trigonometry. 

Plane  Trigonometry. 

Essentials  of  Trigonometry. 

Logarithms  (flexible  covers). 

Elementary  Treatise  on  Logarithms. 


Special  Catalogoie  and  Terms  on  application. 


COLLEGE    ALGEBRA. 


WEBSTER   WELLS,    S.B., 

Pkopessob  op  Mathematics  in  the  Massachusetts 
Institute  of  Technolort. 


LEACH,    SHEWELL,   AND    SANBORN. 

BOSTON.       NEW  YORK.     .CHICAGO. 


COPTRIGHT,  1890, 

WEBSTER  WELLS. 


NottoDoti  ^rcBS : 

J.  S.  Gushing  &  Co.  — Berwick  &  Smith. 

Norwood,  Mass.,  U.S.A. 


:^A 


5^3 


PEEFAOE. 


This  work  is  designed  as  a  text-book  for  the  use  of 
colleges  and  scientific  schools. 

The  first  eighteen  chapters  have  been  arranged  with  ref- 
erence to  the  needs  of  those  who  wish  to  make  a  revieiv  oi: 
that  portion  of  Algebra  preceding  Quadratics.  While  com- 
plete as  regards  the  theoretical  parts  of  the  subject,  only 
just  enough  examples  are  given  to  furnish  a  rapid  review 
in  the  class-room. 

Attention  is  respectfully  invited  to  the  following : 

The  proofs  of  the  five  fundamental  laws  of  Algebra  — 
the  Commutative  and  Associative  Laws  for  Addition  and 
Multiplication,  and  the  Distributive  Law  for  Multiplication 
—  for  positive  or  negative  integers,  and  positive  or  nega- 
tive fractions,  Chapter  II. ;  Arts.  114  and  115 ;  Arts.  208 
to  215 ;  Arts.  230  and  232 ;  Chapter  -XVI. ;  the  proofs  of 
the  fundamental  laws  of  Algebra  for  irrational  numbers, 
Chapter  XVII. ;  Arts.  350  and  351;  Arts.  355  to  357;  Chap- 
ters XXIV.  and  XXVI. ;  the  proof  of  the  Binomial  Theorem 
for  positive  integral  exponents,  Arts.  443  and  444 ;  Chapter 
XXXI. ;  the  Note  to  Art.  469 ;  the  proof  of  the  Binomial 
Theorem  for  fractional  and  negative  exponents,  Art.  483 ; 
Arts.  532  to  538 ;  Art.  542 ;  Chapters  XXXVIL,  XXXVIIL, 
and  XL.;  Art.  650;  the  proof  of  Descartes'  Eule  of  Signs 
for  Positive  Roots,  for  incomplete  as  well  as  complete 
equations,  Art.  653 ;  Arts.  657  to  663 ;  Arts.  673  and  674 ; 


iv  PREFACE. 

the  Graphical  Eepresentation  of  functions,  Arts.  682  to 
688;  Art.  689;  the  solution  of  Cubic  and  Biquadratic 
Equations,  Arts.  706  to  716 ;  Art.  718. 

In  Appendix  I.  will  be  found  graphical  demonstrations 
of  the  fundamental  laws  of  Algebra  for  pure  imaginary  and 
complex  numbers ;  and  in  Appendix  II.,  Cauchy's  proof  that 
every  equation  has  a  root. 

WEBSTER  WELLS. 

Boston,  1890. 


OONTEISTTS. 


PAGE 

I.     Definitions  and  Notation 1 

II.     Fundamental  Operations 9 

III.  Addition;  Subtraction;  Use  of  Parentheses  24 

IV.  Multiplication 29 

V.     Division 36 

VI.     Formulae 41 

VII.     Factoring 50 

VIII.     Highest  Common  Factor    . 59 

IX.     Lowest  Common  Multiple 66 

X.     Fractions 70 

XI.     Simple  Equations  containing  one   Unknown 

Quantity 85 

XII.     Simple    Equations    containing    two   or   more 

Unknown  Quantities 98 

XIII.  Discussion  of  Simple  Equations 113 

XIV.  Inequalities 124 

XV.     Involution 130 

XVI-     Evolution 136 

XVII      Surds;  the  Theory  of  Exponents      ....  1G4 

XVIIL     Imaginary  Numbers lOO 

XIX.     Quadratic  Equations 203 

XX.     Theory  of  Quadratic  Equations 221 

XXL     Problems  involving  Quadratic  Equations     .  234 

XXII.     Equations  Solved  like  Quadratics   ....  239 

XXIII.  Simultaneous     Equations     involving     Quad- 

ratics        246 

XXIV.  Indeterminate  Equations  of  the  First  Degree  262 

V 


vi  CONTENTS. 

^  PAGE 

XXV.     Ratio  and  Proportion 268 

XXVI.     Variation 279 

XXVII.     Arithmetical  Progression 285 

XXVIII.     Geometrical  Progression 295 

XXIX.     Harmonical  Progression 305 

XXX.     The    Binomial    Theorem;    Positive    Inte- 
gral Exponent 309 

XXXI.      CONVERGENCY   AND   DIVERGENCY   OF    SeRIES    .  316 

XXXII.     The   Theorem    of    Undetermined    Coeffi- 
cients       328 

XXXIII.  The  Binomial   Theorem  ;   Fractional  and 

Negative  Exponents 344 

XXXIV.  Logarithms 352 

XXXV.     Compound  Interest  and  Annuities    .     .     .  378 

XXXVI.     Permutations  and  Combinations  ....  385 

XXXVII.     Probability  (Chance) 393 

XXXVIII.    Continued  Fractions 407 

XXXIX.     Summation  of  Series 418 

XL.     Determinants 429 

XLL     Theory  of  Equations 450 

XLII.     Solution  of  Higher  Equations      ....  498 


APPENDIX  I.  Demonstration  of  the  Fundamental 
Laws  of  Algebra  for  Pure  Imagi- 
nary and  Complex  Numbers   .     .     .     529 

APPENDIX  II.  Cauchy's  Proof  that  every  Equa- 
tion HAS  A  Root 54(1 


ANSWERS  TO  THE  EXAMPLES. 


QUADRATIC   EQUATIONS.  20^ 


XIX.    QUADRATIC  EQUATIONS. 

339.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree  (Art.  179),  containing  but  one  unknown  quantity. 

A  Pure  Quadratic  Equation  is  a  quadratic  equation  involv- 
ing only  the  square  of  the  unknown  quantity  ;  as,  2a^=  5. 

An  Affected  Quadratic  Equation  is  a  quadratic  equation 
involving  both  the  square  and  the  first  power  of  the  un- 
known quantity  ;  as,  2  a-^  —  3  a;  —  5  =  0. 

PURE    QUADRATIC  EQUATIONS. 

340.  A  pure  quadratic  equation  may  be  solved  by  reduc- 
ing it,  if  necessary,  to  the  form  x^  =  a,  and  then  extracting 
the  square  root  of  both  members. 

1.  Solve  the  equation       Sa.-^  -f-  7  =  — -  +  35. 

Clearing  of  fractions,     12  x-  -f  28  =  5  a;^  + 140. 

Transposing  and  uniting,         7  a.-^  =  112, 
or,  a.-2  =  16  ; 

which  is  in  the  form  x^  =  a. 

Extracting  the  square  root  of  both  members,  we  have 

a;  =  ±  4.  (:N"otes  1  and  2.) 

Note  1.  The  sign  ±  is  placed  before  the  result,  because  the  square 
root  of  a  number  is  either  positive  or  negative  (Art.  236).  ■ 

Note  2.  It  follows  from  Art.  267,  Note,  that  hke  roots  of  the  mem- 
bers of  an  equation  are  equal. 

2.  Solve  the  equation       7a^  —  5  =  5.^'^  —  13. 
Transposing  and  uniting,         2  a;' =  —  8, 

or,  a^  =  —  4  j 

which  is  in  the  form  x^  =  a. 


204  COLLEGE   ALGEBRA. 

Whence,  x  =  ±-\/— 4 

=  ±2V^^   (Art.  332). 

Note  3.  In  this  case  the  values  of  x  are  imaginary  (Art.  326)  ;  it 
is  impossible  to  find  any  real  values  of  x  which  will  satisfy  the  given 
equation. 

EXAMPLES. 

341.    Solve  the  following  equations  : 

J     _5 L  =  _^.  3     _A_  =  § ^. 

■    6ic2      4a^  16*  ■    4  — a;      3      4  +  a;' 

h 


2.    4-V3ar'  +  lG  =  6.  4. 


b      XT  —  a 

5.  2(a;  +  3)(a;-3)  =  (a;  +  l)2-2a;. 

6.  (3a;  -  2)  (2x  +  5)  +  (5a;  +  1)  (4x  -  3)-  91  =  0, 

^-    2~"  +  l2-24~^+^' 

g     2a;^-5  _  3«Mi_2  _  x"  -  10  ^  ^ 
3  7  6* 

^  *~^^  10.    2 ^^ —  =  -/'3 


x  +  b      a  — 2b  2;ir  —  1      3  \        3.x-^'  +  2 

11.    (2 X  -  a)  (a;  +  6)  +(2 a;  +  a)  {x-b)  =  cr  +  b'. 

12     5a.-^-l      3a;^  +  l 89  ^^ 

a-2-3        x-2  +  2        (.r^  -  3)  (ar^  +  2) 

13.    X  -\-  Var  4-  3  =        ^ 

Var^  +  3 

14  1  1  _V3 


15.    Vl  +  a;  +  a;-  +  Vl  —  a;  +  x-^  =  V7  +  V3. 


16. 


M-l  ,      |a;=^_2         R'-3  ,      Ix" - 


-4 
3' 


■V 


QUADRATIC   EQUATIONS.  20^ 


AFFECTED   QUADRATIC   EQUATIONS. 

342.  By  transposing  all  terms  containing  the  unknown 
quantity  to  the  first  member,  and  all  other  terms  to  the 
second  member,  any  affected  quadratic  equation  may  be 
reduced  to  the  form 

ax^  -\-hx  =  c. 

343.  To  solve  a  quadratic  equation  of  the  form 

a!(?  -{-hx  =  c. 

If  the  coefficient  of  x"^  is  a  perfect  square,  the  equation 
may  be  solved  by  adding  to  both  members  such  an  expres- 
sion as  will  make  the  first  member  a  perfect  trinomial 
square,  an  operation  which  is  termed  completing  the  square, 
and  then  extracting  the  square  root  of  both  members. 

1.    Solve  the  equation    9ar^  +  2  a;  =  11. 

A  trinomial  is  a  perfect  square  when  its  first  and  third 
terms  are  perfect  squares  and  positive,  and  the  second  term 
plus  or  minus  twice  the  product  of  their  square  roots  (Art. 
125). 

Therefore  the  square  root  of  the  third  term  is  equal  to 
the  second  term  divided  by  twice  the  square  root  of  the 
first. 

Hence  the  square  root  of  the  expression  which  must  be 

2x       1 
added  to  9  x^  +  2  x  to  make  it  a  perfect  square,  is  — ,  or  -• 

-J  ox       o 

Adding  to  both  members  the  square  of  -,  we  have 


Extracting  the  square  root  of  both  members  (Art.  126), 
(See  Art.  340,  Notes  1  ami  2.) 


^"^  +  3"*  3 


206  COLLEGE   ALGEBRA, 

o  1^10      o  11 

Whence,  x  =  1  or  — —  • 

We  then  have  the  following  rule  : 

Complete  the  square  by  adding  to  both  members  the  square 
of  the  quotient  obtained  by  dividing  the  coefficient  of  x  by  twice 
the  square  root  of  the  coefficient  of  a^. 

Extract  the  square  root  of  both  members. 

Note.   The  values  of  x  may  be  verified  (Art.  175)  as  follows : 

Putting  a;  =  1  in  the  given  equation,     9  +  2=11. 

~  =  -^.  i|l_|=n. 

If  the  coefficient  of  x^  is  unity,  the  rule  may  be  modified 
to  read  as  follows  : 

Complete  the  square  by  adding  to  both  members  the  square 
of  half  the  coefficient  of  x. 

2.  Solve  the  equation   cc^  —  llx  =  —  2S. 

Adding  to  both  members  the  square  of  — j 

^_n.  +  (f)=-28  +  f=2. 

Extracting  the  square  root, 

11       ,3 

^  — ^  =  ±  ^• 
2  2 

Whente,  a;  =  ii  ±  ^  =  7  or  4. 

2       2 

344.  If  the  coefficient  of  a^  is  not  a  perfect  square,  it 
may  be  made  so  by  multiplication. 

3.  Solve  the  equation   3  a;^  +  13  tc  =  —  12. 

The  coefficient  of  a^  may  be  made  a  perfect  square  by 
multiplying  each  term  by  3  ;  thus, 


QUADRATIC   EQUATIONS.  207 

9a^  +  39x  =  -36. 
Completing  the  square  by  the  rule  of  Art.  343, 

Extracting  the  square  root, 

2  2 

3a;  =  -— ±-  =  -4or  -9. 

2      2 

Whence,  x-= or  —  3. 

3 

If  the  coefficient  of  v?  is  negative,  the  sign  of  each  term 
must  be  changed. 

4.    Solve  the  equation    —%x^-\-15x  =  —  2. 

Changing  the  sign  of  each  term,  we  have 

8ar-15rc  =  2. 

The  coefficient  of  a?  may  be  made  a  perfect  square  by 
multiplying  each  term  by  2  ;  thus, 

16  ar'- 30  a;  =  4. 

Completing  the  square, 

'  4  ^         ^16       16 


Extracting 

the 

square 
4a; 

root, 

15  _ 
4 

-?■ 

4x  = 

f* 

17 
4 

=  8 

or 

1 
2 

Whence, 

X  = 

:2  or 

- 

1 

208  COLLEGE  ALGEBRA. 

345.   Fractions  may  be  avoided  in  completing  the  square 
by  multiplying  both  members  of  the  given  equation  by  four 
times  the  coefficient  of  a?. 
For  consider  the  equation 

ax^  -\-  hx  =  c. 
Multiplying  both  members  by  4  a,  we  have 

4  a^a^  +  4  dbx  =  4  ac. 
Completing  the  square  by  the  rule  of  Art.  343, 

4  aV  +  4.abx  +  b'  =  b''-i-A  ac. 
Extracting  the  square  root, 


2ax  +  &  =  ±  V&^  +  4ac. 
2ax  =  —  b±  Vb^  +  4  ac. 


Whence,  ^^-b±^b'  +  4ac 

2a 

It  will  be  observed  in  the  above  case  that  the  quantity 
required  to  complete  the  square  is  the  square  of  the  coeffi- 
cient  of  x  in  the  given  equation. 

.   5.    Solve  the  equation   2a^  —  3x=:lA. 

Multiplying  both  members  by  4  times  2,  or  8, 

16ar^-24a;=112. 
Adding  to  each  member  the  square  of  3, 

Wx"  -  24a;  +  3^  =  112  +  9  =  121. 
4a; -3  =  ±11. 

4a;  =  3±  ll  =  14or  -8. 

Whence,  x  =  ~  or  —  2. 

2 

If  the  coefficient  of  x  in  the  given  equation  is  even,  the 
rule  may  be  modified  to  read  as  follows  : 

Multi2^ly  both  members  of  the  equation  by  the  coefficient  of 
a?,  and  add  to  each  the  square  of  half  the  coefficient  of  x  in  the 
given  equation. 


QUADRATIC    EQUATIOJ^S.  209 

6.    Solve  the  equation    9  a;-  +  14  a;  =  —  1. 

Multiplying  both  members  by  9, 
81x-  +  12G.T  =  -9. 
Adding  to  each  member  the  square  of  7, 

81x2  +  126x  +  72  =  -  9  +  49  =  49. 

9a;  +  7  =  ±V40  =  ±2VlO  (Art.  295). 
9a;  =  -7±2VlO. 

Axn.                                                -  ^  ±  2  VlO 
Whence,  x  = 

Note.   The  method  of  complethig  the  square  exempUfied  in  the 
present  article,  is  known  as  tlie  Hindoo  Method. 

EXAMPLES. 

346.  Solve  the  following  equations  : 

1.  4.x'  +  3x  =  10.  7.  11  a;  +  12- 36 .^•-  =  0. 

2.  x^  —  x  =  6.  8.  6.^■2-5a;  =  -l. 

3.  25£c2-15a;  =  -2.  9.  9x'  +  6x=19. 

4.  49i»2  +  49x  +  10  =  0.  10.  32rc2^  20a;  -  7  =  0. 

5.  2a;2_i5^^_13  n  32 a; -  48 o.-^  =  - 3. 

6.  8a;2  +  a;-34  =  0.  12.  12ar'+ 5a; +  1  =  0. 

SOLUTION  OF  QUADRATIC  EQUATIONS  BY  A  FORMULA. 

347.  It  was  shown  in  Art.  345  that  if 

ax^  -\-  bx  =  c, 


,,                                                      —b±  Vb^  +  4  ac  ...  X 

then  x  =  - "^ — ^ (1) 

2  a 

This  result  may  be  used  as  ^  for  mi  da  for  the  solution  of 
any  quadratic  equation  in  the  form  ax^  +  bx  =  c. 


210  COLLEGE   ALGEBRA. 

1 .  Solve  the  equation   2  a^  +  5  ic  =  18. 

Here,  a  =  2,  b  =  5,  and  c  =  18;  substituting  in  (1), 

-  5  ±  V25  +  144  _  -  5  ±  VT69 
^-  4  4 

^-5^13^2or-5. 
4  2 

2.  Solve  the  equation   110  a^  —  21  cc  =  —  1. 
Here,  a  =  110,  &  =  —  21,  and  c  =  —  1 ;  therefore, 


21  ±  V441  -  440      21  ±1       1         1 

X  = = =  —  or  — 

220  220        10       11 

Dividing  both  terms  of  the  fraction  in  equation  (1)  by  2, 
we  have  by  Art.  297, 

x  = = ^ ;  (^) 

a  a 

which  is  a  convenient  formula  to  use  in  case  the  coefficient 
of  X  in  the  given  equation  is  even. 

3.    Solve  the  equation    —  Sx^  +  14  a;  =  —  3. 

Here,  a  =  —  5,  6  =  14,  and  c  =  —  3 ;  substituting  in  (2), 


^^_7±V49  +  15^-T±8^_1^^3_ 


EXAMPLES. 
Solve  the  following  equations  : 

4.  2x-'  +  3a;  =  27.  8.    6a;2  +  7a;  =  -l. 

5.  3ar'-2a;  =  5.  9.    4x2_8aj_3  =  o. 

6.  x2-7x  =  -10.  10.    5x^  +  12a;  =  -4. 

7.  5x'-^  +  x  =  18.  11.    0x2 -25a; +  14  =  0. 


QUADRATIC   EQUATIONS.  211 

12.  S0a;-9a;2  =  16.  14.    15a;2-8a;  =  16. 

13.  a.-2  + 39a; +  387  =  0.  15.    10 -21a;-10a;2  =  0. 

348.  The  following  equations  may  be  solved  by  either  of 
the  preceding  methods,  preference  being  given  to  the  one 
best  adapted  to  the  example  under  consideration. 

1.    Solve  the  equation 

3x         2a;-3^  _5 
2a;-3         3x     ~      6 

The  equation  must  first  be  reduced  to  the  form  aar+6a;=c. 

Multiplying  both  members  by  6  cc (2  a;  —  3), 

ISa^  -  2(2a;  -  3)2  =  -  5a;(2a;  -  3). 
ISar^  -  8a;2  +  24a;  -  18  =  -  10^2  _,.  -^^^ 

20x'-\-9x  =  18, 
which  is  in  the  form  aar  +  6a;  =  c. 

Solving  by  formula  (1)  of  Art.  347,  we  have 


-  9  ±  V81  -{- 1440 

40 
-9±39_3^^ 
40           4 

6 

5' 

The  artifice  employed  in  the  following  example  may  often 
be  used  to  advantage. 

2,   Solve  the  equation 

2ar'-a;  +  3      3a^+a;  +  2^2ar'-2a;  +  3 
2a;-l  3a;  +  l  a;-l 

The  equation  may  be  written  in  the  form 

a;(2a;-l)  +  3      .T(3.'r +1) +  2^2a;(a; -l)  +  3 
2a; -1        "^        S.T  +  1         ~         a;-l 

Dividing  each  numerator  by  the  corresponding  denomi- 
rator. 


^ 


!12  COLLEGE  ALGEBRA. 


Q  9  o 


2a;-l  3a;  +  l  x-1 

Whence,  — ^ h  — - —  =     "^    . 

2x-l      3x+l      a;-l 

Clearing  of  fractions, 

9aj2  _  g^.  _  3  _,_  4^  _  g^  _^  2  =  18a;2  -  3a;  -  3. 

-5ar-9a;  =  -2. 


Whence,  ^.^9±V81+40 

-10 

9  ±  11  ^      1 

= =  —  2  or  — 

-10  5 


EXAMPLES. 

Solve  the  following  equations : 

3    2  +  ?=-:^.  4    2__5^^_15 

'   X     2         2  '   5     2x         4.x'' 

5.  4a;(18a;-l)  =  (lOx-l)l 

6.  {3x-5y-{x  +  2y=-5. 

7.  (x  +  Sy-  {x-iy=19. 

8.  {x-iy-{3x-\-8y-(2x  +  5y  =  0. 


9. 

2a; +  3      2a; +  9      ^ 
8  +  a;       3a;  +  4 

14. 

X         a;  - 1      3 
a;-l         X         2 

^ 

5     3a;  +  l      1 

a;          a;^          4 

15. 

a;         5  -  a;      15 

5  —  a;         a;          4 

11. 

4  a;      14 -a;  ^14 

a;  +  l 

16. 

a-  +  1      a:  +  3      8 
a;  +  2      a;  +  4      3 

12. 

^^^  _  E  _  34  =  0. 

17. 

V20  +  a;-a;2^2a; 

13. 

3ar^        l-8a;      a; 
a; -7          10         5 

18. 

2V:t^+— =  5. 

V.C 

•w. 


19. 
20. 
21. 
22. 

2a. 

24. 
25. 


QUADRATIC   EQUATIONS. 

2a; -1  3x 


213 


3a;-l 


+i  =  o. 


a;3  _  a.-^  ^  7  ^         11 
cc2-{-3x-l      '*^"'~3' 

2ar'  +  3a;-5^3a;^  +  4a;-l 
2x2-a;-l       3x2-2x-  +  7' 

7  3     ^22 

a;2-4      x  +  2       5* 

1  111 

3' 


x^-l      3a;-3      a;  +  l 

3x  +  2      2a;  +  3  ^       2 
CC4-3        a;-3 

12+5a;  1 


9-ar' 
J  2+a; 


=  0. 


12  —  5a:      1  —  5a;         x 
a;  +  l  =  l. 


0. 


26.  V4X--3 

27.  2V^-  Vx  +  5  = 
a;4-2         6 


28 


Va;  +  5 

^2      ^-^. 

a  —  1      a;+5      '^      x  +  l 


29.  V3a;  +  1  =  V9a;  +  4- V2a;-1. 

30.  (a;-3)(a;  +  4)  =  (a;-7)V3. 

31  ^  —  ^  _  a;  — 4  _  a;  +  l  _  a;  +  2 
x  —  3     x  —  5     x-\-4:     x  +  5 

32  gar' +  4a;     4a;^  +  8a;  +  5      3a:^-x 


3a;  +  2 


2a;  +  3 


=  0. 


33. 


+  1  + Va^_l_9(a;.t.l) 


a;  +  l^Va;2-l  8 

34.    (11  4-  6V2)x'-\-  (1 V2  -  \))x  =  7 V2  -  10. 


214  COLLEGE   ALGEBRA. 

SOLUTION  OF  LITERAL  QUADRATIC  EQUATIONS. 

349.  For  the  solution  of  literal  affected  quadratics,  the 
methods  of  Art.  345  will  be  found  in  general  the  most  con- 
venient. 

1.  Solve  the  equation 

x^  +  ax  —  bx  —  ab  =  0. 

The  equation  may  be  written 

ar'  +  (a  —  b)x  =  ab. 

Multiplying  both  members  by  4  times  the  coefficient  of  x', 

4ar^  +  4(a  —  6)a;  =  4a6. 

Adding  to  each  member  the  square  of  a  —  6, 

4.ari-{.  4:(a  -  b)x  +  {a  -  by  =  Aab  +  a"  -  2ab  +  b'' 

=  a'  +  2ab  +  b\ 

Extracting  the  square  root, 

2x-\-(a-b)=  ±(a  +  b). 

2x=  -{a-b)±(a  +  b). 

Therefore,  2x=-a  +  b  +  a  +  b  =  2b, 

or  2x=  —a-^b  — a  — b= —2a. 

Whence,  x  =  b  or  —a. 

Note.  If  several  terms  contain  the  same  power  of  x,  the  coefficient 
of  that  power  should  be  enclosed  in  a  parenthesis,  as  shown  above. 

2.  Solve  the  equation 

(m  —  l)a^  —  2m^cc  =  —  4m^ 
Multiplying  both  members  by  ??i  —  1, 

(m  -  lyx""  -  2w?{m  -  l)x  =  -4.m\m  -  1). 
Adding  to  each  member  the  square  of  w?, 

{m-iyx^-2m-(^,n-l)x  +  vi'=  vi*-  4mH  4m*. 


QUADRATIC   EQUATIONS.  215 

Extracting  the  square  root, 

{m  —  l)x  —  m,^=  ±(m-  — 2m). 

(m  —  l)x  =  m^-\-m^—2vioTm"—  m--\-2m 
=  2m(m—  1)  or  2m. 

Whence,  a;  =  2  m  or  -1^^. 

m  —  1 

EXAMPLES. 

Solve  the  following  equations  : 
,    3.   ar'  +  2ma:=2m  +  l.  7.   0:^  +  2(0  +  8)0;  = -32c. 

4.  u?-2ax={h-\-a){h-a).       8.   a^  -  m^a:  +  m^'a;  =  m^ 

5.  a?  —  ax  +  hx  —  ab=2  0.  9.   acx^ —  bcx  —  adx  =  —bd 
.   6.   cc2  +  (a  +  l)a;= -a.            10.    (x+22)y-(x+py=37p' 

11.   6ar^-h9ax-{-2bx  +  3ab  =  0. 

12.    2a;(a-a;)^a^  ^^    a:  +  l  =  ^  +  ^. 

3a  —  2a;       4  a;      w      m 

13    _^  =  _^.  15.   ^-±1  =  ^L±1. 

'   a;  +  l      ?i  +  l  Vx         Va 

16.    V(a  +  6)a; -  4a&  =  a;  - 2&. 

17.  v^^r4;^=(^  +  ^>(«-^). 

\18.   2V^3^  +  3V2^  =  '^^  +  ^^.  > 
Va;  — m 

19.   ^=  + L^  =  l  +  ?. 

a  +  Va^  —  a;     a  —  V  a^  —  a;  « 


ar'  +  l^g  +  S         c_ 
X  c         a  +  b 

a;  —  a      x  —  b      a      b 


V  20. 


21(3  COLLEGE  ALGEBRA. 

22     a^  +  m  ^   x-\-n  ^5 
X  +  ?i      a;  +  ??i      2 

oo     cc  —  a.x  +  a      5a.'c  —  3a  —  2      ^ 

<*«5 . 1 =  U. 

x+  a      X  —  a  a-  —  X- 

24.  _^  =  1  +  1  +  1. 

a  +  b  -\-x      a      b      X 

25.  x{x-\-b  +  c)  =  {a-\-c){a-b). 

26.  (c  +  a  -  2&)a;2  -|-(a  +  6-2c)a;-+-6  +  c-2a  =  0. 

27.  a^^  I  3a''a;^6a^  +  a6-26^      6^x 

c  c?  c 

28.  (3a2  +  62)  (a:2  _  ^  ^  1)  ^  (3^2  j^  a"")  {:i?  +  x  + 1). 

29.  (2 a^  +  a6) a^ -  (7  a^  +  11  a6  +  4 62) a;  +  3  a^  +  4 a&  =  0. 

30    ^  —  O'  —  b  —  c     x  —  a  +  b  +  G  _  _  6^  +  c^ 
X  —  a  +  b  +  c     X  —  a  —  b  —  c  be 


31 .  (a  +  b)  Va^  +  6^  +  X  -  (a  -  &)  Va^  +  6^  _  a;  =  «=>  +  61 

32.  (a?  +  l){a*  +  a'  +  l)  =  2x{a*  +  3a'  +  l). 
33     c[^  +  ^(g  +  l)  _  x  +  b  _ 


ax  —  b(a-j-l)         X         2ax  —  b 

34.  {6a'  +  llab-10b')x'-(ldd''+9ab  +  13b')x 

=  -15a'  +  17ab  +  Ab\ 

35.  (a" -b^-c'-2bc) {x"  + 1)  =  2a;(a2  +  62  +  0^  +  26c) 


SOLUTION  OF  EQUATIONS  BY  FACTORING. 

350.   Let  it  be  required  to  solve  the  equation 

(x-S){2x  +  5)  =  0. 

It  is  evident  that  the  equation  will  be  satisfied  when  x 
has  such  a  value  that  one  of  the  factors  of  the  first  member 
is  equal  to  zero ;  for  if  any  factor  of  a  product  is  equal  to 
zero,  the  product  is  equal  to  zero. 


QUADRATIC   EQUATIONS.  217 

Hence,  the  equation  will  be  satisfied  when  x  has  such 
a  value  that  either 

a;  -  3  =  0,  (1) 

or  2a; +  5  =  0.  (2) 

K 

Solving  (1)  and  (2),  we  have  a;  =  3  or 

z 

K 

Therefore  the  roots  of  the  given  equation  are  3  and 

In  general,  if  A,  B,  ...,  K  are  any  integral  expressions 
(Art.  65)  containing  x,  the  equation 

^•x5x  •••  x/r=0 

may  be  solved  by  placing  the  factors  of  the  first  member 
separately  equal  to  zero,  and  solving  the  resulting  equations. 


EXAMPLES. 
351.    1.  Solve  the  equation  4a;^  —  2  a;  =  0. 
Factoring  the  first  member, 

2a;(2a;-i)  =  0. 
Placing  the  factors  separately  equal  to  zero  (Art.  350), 
2a;  =  0,  or  a;  =  0  ; 

and  2.T  — 1  =  0,  or  x  =  — 

2 

Therefore  the  roots  of  the  given  equation  are  0  and  — 

A 

2.    Solve  the  equation  x'^-|-4a;2  —  a;  —  4  =  0. 
Factoring  the  first  member  (Art.  120), 

ix-\-^){7?-\)  =  ^. 
Therefore,  a;  -}-  4  =  0,  or  a;  =  —  4 ; 

and  a;^  —  1  =  0,  or  a;  =  ±  1. 


218  COLLEGE   ALGEBRA. 

3.    Solve  the  equation  a:^  —  1  =  0. 
Factoring  tlie  first  member, 

Therefore,  x  —  1=0,otx  =  1; 

and  a^  +  x  +  l  =  0. 

Whence,  by  Art.  347,   x  =  -^^Vl^^ ^  -l±V-3^ 

Z  L 

Note.  The  above  examples  are  illustrations  of  the  principle  (Art. 
634)  that  the  degree  of  an  equation  (Art.  179)  indicates  the  number  of 
its  roots ;  thus,  an  equation  of  the  third  degree  has  three  roots ;  an 
equation  of  the  fourth  degree,  four  roots  ;  etc. 

It  should  be  observed  that  the  roots  are  not  necessarily  unequal ; 
thus,  the  equation  x^  —  2x  +  l  =  0  may  be  -written  (x  — 1)  (x  —  1)  =  0, 
and  therefore  the  two  roots  are  1  and  1. 

Solve  the  following  equations  : 
4.    (Zx  +  l){^a?-4Q)  =  0.       6.    3:x?  +  l^a?  =  0. 
6.   2a:3_i8a;  =  0.  7.    {x" -^){x' +  4.)  =  0, 

8.  a?-\-^x'-24.x  =  Q. 

9.  {x-2){2a?  +  13x+2Q)  =  Q. 

10.  24.x' -2a? -127?  =  0.        12.   x3_27=:0.   - 

11.  a;*  +  x  =  0.  13.    a;*-16  =  0. 

14.  (a;2-a2)(4a^-4aa;-15a2)  =  0. 

15.  (x2-5cc  +  6)(ic2+7a;  +  l2)(ar'-3a;-4)  =  0. 
16.   0^-1  =  0.  -   18.   27x3 +  640=^  =  0. 
17.\8a;3^125  =  0.                 "- 19.   x^-a^.g^j^.  9  =  0. 

20.'^2a;3  -  3x5+ 2x  -  3  c=  0. 
21.   3x*  +  x*-3Gx-2-12x  =  0. 


QUADRATIC   EQUATIONS.  219 


22.    Solve  the  equation  V2  —  3  a;  +  Vl  -f  4  a;  =  V 3  +  a;. 
Squaring  both  members, 


2  -  3a;  +  2V2  -  3a;vT+4^  +  l  +  4a;=  3 +05, 


Whence,  2  V2  -  3  xVl  +  4  a;  =  0 ; 

or,  V2-3a;Vl+4aj  =  0. 

Squaring,  (2  -  3  a;)  (1  +  4  a;)  =  0. 

2 
Therefore,  2  — 3a;  =  0,  or  x  =  ~; 

o 

and  1  4-4a;  =  0,  or  a;  =  — -• 


2 

23.    Solve  the  equation 


a;  +  8      x  +  9      a;  +  10      a;  +  6 
Adding  the  fractions  in  each  member,  we  have 

7a; +  58        ^        7a; +  58  • 

(a;  +  8)(a;+9)      (a;  + 10)  (a;  +  6) ' 

Since  7a;  +  58  is  a  factor  of  each  member,  we  may  place 
it  equal  to  zero ;  thus, 

7a;  +  58  =  0,  or  a;  =  -— • 

The  remaining  roat  is  given  by 

1  1 


{x  +  8)  (a;  +  9)       (a;  + 10)  (a;  +  6) 

or,  (a;  +  8)  (a;  +  9)  =  (a;  + 10)  (a;  +  6). 

That  is,       af-hnx  +  72  =  x^  +  16x-\-60, 

or,  a;  =  — 12. 

Therefore  the  roots  of  the  given  equation  are  — 12  and 
_58 
7' 


220  COLLEGE  ALGEBRA. 

Solve  the  following  equations  ; 

24.  Vx  +  a  +  Vxl^b  =  V2ic  +  a  +  6,      v 

25.  -±-  +  -^—  +  -^—  =  3. 

a;  +  1      a-  +  2  .    .r  +  3 

26     ^~^'  =  a;^-4ar^  +  9 
'    2  +  x^      ar'  +  4a^  +  9' 


27.    V2  —  3  cc  -  V7  +  a;  =  V5  +  4a;.      V 


28.    V(2  +  a;)(a;-f-l)+V(2-a;)(a;-l)=  V6^ 


a; +8      a;  — 8      x  +  6      x  —  6 
30.    Vo/*^  +  aa;  —  6  a-  —  Va;^  —  ax  —  6  a^  =  V^a;-—  12  a^. 


31.    V4  +  ox  —  a.-2  =  2V2a;  +  Va;-  +  3a;  —  4. 

32     x^+5^'+l      2x-^-2x  +  l      3x^-15.^  +  1  ^  6x^+6x--?. 
x  +  5  X— 1  X—  5  x  +  1. 

1.1.1.1 


=  0. 


x-\-a  +  b'x  —  a  —  b      x  +  a  —  b      x  —  a  +  b 


36.    Vx^  -  8x  +  15  +  Vx-'  4-  2x  -  15  =  V4x^  -  18x  +  18. 

36.*    (a-xy  +  (b-xy  =  {a  +  b-2x)\ 

3^     x^-K2x  +  4^  4 
x^  —  2  X  -f-  4      x^ 

38.    (3x  +  2)3-(2x-l)3  =  (x+3)l 

*  Factor  the  first  member  by  the  method  of  Art.  130. 


THEORY  OF   QUADRATIC   EQUATIONS.         2-21 


XX.  THEORY  OF  QUADRATIC  EQUATIONS. 

352.  A  quadratic  equation  cannot  have  more  than  two 
different  roots. 

Every  quadratic  equation  can  be  reduced  to  the  form 

3!^  -\-px  =  q. 

If  possible,  let  this  equation  have  three  different  roots, 
Vi,  rg,  and  r^. 

Then  by  Art.  174,  r^^  +2^n  =  Q>  (1) 

r/  +i)?-2  =  q,  (2) 

and  ri-\-pr^  =  q.  (3) 

Subtracting  (2)  from  (1),  we  have 

I"!  —  t'i  +  i>  (^1  —  rg)  =  0. 

Whence,  (ri  +  r^  (rj  —  rj)  +  p  (rj  —  r^ )  =  0, 
or,  (ri  -  rj)  {r^  +  r2  +  i?)  =  0. 

Therefore,  either  r^  —  rg  =  0,  or  else  ^i  +  rg  +  i>  =  0. 

But  ri  —  ?*2  cannot  be  zero,  for  by  hypothesis  r^  and  r, 

n  +  ^-2  +P  =  0.  (4) 

In  like  manner,  by  subtracting  (3)  from  (1),  we  obtain 

^i+r3+iJ  =  0.  (5) 

Subtracting  (5)  from  (4),  rg  —  rg  =  0. 

But  this  is  impossible,  for  by  hypothesis  r^,  and  r^  are 
different;  hence  a  quadratic  equation  cannot  have  more 
than  two  different  roots. 

Note.  It  follows  from  the  above  that  an  expression  cannot  have 
more  than  two  different  square  roots. 


22  COLLEGE  ALGEBRA. 

353.   Let  Ti  and  rg  denote  the  roots  of  the  equation 

a:r  -\-px  =  q. 
By  Art.  347,  r,  =  -P+^P'  +  ^q, 


(1) 


and  ^^^-P-^P'  +  ^Q.  (2) 

Adding  (1)  and  (2), 


Multiplying  (1)  and  (2),  we  have 
P'-i 

-4g 


^^,^^f-(p'^  +  ^g)     (Art.  108) 


4     =-^- 

Hence,  if  a  quadratic  equation  is  in  the  form  x^+pa;  =  q, 
the  sum  of  the  roots  is  equal  to  the  coefficient  of  x  with  its  sign 
changed,  and  the  product  of  the  roots  is  equal  to  the  second 
member  with  its  sign  changed. 

EXAMPLES. 

1.  Find  the  sum  and  product  of  the  roots  of  the  equa- 
tion 2a:2_7a._i5^0. 

Transposing  — 15,  and  dividing  by  2,  the  equation  becomes 
o      7x      15 

7                                     15 
Hence,  the  sum  of  the  roots  is  -  and  their  product 

Find  by  inspection  the  sum  and  product  of  the  roots  of : 

2.  x='  +  5a;  +  2  =  0.  6.   8x'-x  +  4.  =  0. 

3.  a^-7x  +  ll  =  0.  7.    6a;-4ar'  +  3  =  0. 

4.  x'  +  c>x-l  =  0.  8.    7  -12x-Ux'  =  0. 

5.  22ty'-3x-2  =  0.  9.    4x2- 4ax  +  a^  -  6- =  0. 


THEORY  OF  QUADRATIC  EQUATIONS.    223 

10.    If  rj  and  rj  are  the  roots  of  the  equation  x^  -f-^jx  =  ^, 

(a.)    Prove  that      1  +  1=-^. 

(6.)    Prove  that    r/  +  ri  =p-  +  2q. 

354.  By  aid  of  the  principles  of  Art.  353,  a  quadratic  equa- 
tion may  be  formed  which  shall  have  any  required  roots. 

For,  let  ri  and  r^  denote  the  roots  of  the  equation 

x'+px-q^O.  (1) 

Then  by  Art.  353,  p  =  —  (ri  +  r^)  and  —q  =  nr.^. 
Substituting  these  values  in  (1),  we  have 
a^—  (^1  +  ^2)^  +  '"1^2  =  0, 
or,  {x  —  ri)  (x  —  r2)  =  0. 

That  is,  any  quadratic  equation  may  be  written  in  the 

form 

{x-r,)(x-n)  =  0,  (2) 

where  Ti  and  rj  are  its  roots. 

Hence,  to  form  a  quadratic  equation  which  shall  have 
any  required  roots. 

Subtract  each  of  the  roots  from  x,  and  place  the  p)'roduct  of 
the  resulting  expressions  equal  to  zero. 

EXAMPLES. 
1.    Form  the  quadratic  equation  whose  roots  shall  be  4 

and 

4 

By  the  rule,  (x  -  4)  (  a;  +  ^  )  =  0. 

Multiplying  by  4,  (x  -  4)  (4a;  +  7)  =  0. 
That  is,  4  ar'  -  9  x  -  28  =  0. 


224  COLLEGE  ALGEBRA. 

Form  the  quadratic  equations  whose  roots  shall  be  -. 
2.  4,  5,         4.  3,  _|.  6.  I  I         8.    -f,  0. 

3.,. 3.     a.-.-f.      7.    -|i     9.    -I  4 

10.  a-6,  a  +  2&.  12.   2  +  5 VS,  2  -  5 Va 

11.  ?n(l  +  m),  m(l-m).         13. ?    ^ 


FACTORINQ. 

355.  A  quadratic  expression  is  an  expression  of  the  form 
aoi?  -\-hx  •\-  c. 

The  principles  of  Art.  354  serve  to  resolve  such  an  ex- 
pression into  two  factors,  each  of  the  first  degree  in  x. 

We  have  aar^ +  6x +  c  =  a^ar +  -|  +  ^Y  (D 

Now  let  rj  and  rg  denote  the  roots  of  the  equation 

a?  +  *^  +  °=0. 
^  a      a 

By  Art.  354,  (2),  the  equation  can  be  written  in  the  form 
(x  —  rj)  (x  —  Ti)  =  0. 

Hence,  the  expression  a? -\ '  -\-~  can  be  written  in  the 

form  {x  —  ri)  (x  —  r^) . 

Substituting  in  (1) ,  we  have 

aa?  +  hx  +  c  =  a{x  —  r-^){x-ri),  (2) 

where  rj  and  r^  are  the  roots  of  the  equation  ar^+  —  +  -  =  0, 

or    aoi?  +  hx  +  c  =  0\    which,  we   observe,   is   obtained    by 
placing  the  given  expression  equal  to  zero. 


THEORY  OF  QUADRATIC  EQUATIONS.    225 

EXAMPLES. 
1.   Factor  6  0^  +  7  a; -3. 
Placing  the  expression  equal  to  zero,  we  have 

Whence  by  Art.  347, 


^  ^  -  7  ±  V49  +  72^-7±ll^l       _3 
12  12  3  °^      2* 

1  3 

Here,  a  =  6,  rj  =  -,  rg  =  —  - ;  substituting  in  (2), 
o  Z 

=  (3a;-l)(2a;  +  3), 

2.   Factor  4  + 13  a;- 12  a:2_ 

Solving  the  equation  4  +  13  k  —  12  a;^  =  0,  we  hare 

a.^-13±Vl<39  +  li)2^ -13±19^      1       4 
_24  -24  4°^  3* 

Whence,        4  +  13a;  -  12a.-2  =  - 12  ("a;  +  -)  fx  -  ^"j 

=  (1  + 4a;)  (4 -3a;). 

Note,  It  must  be  reraembered  that  a,  in  the  formula  a(x—r{)  («— rj"), 
represents  the  coefficient  of  x-  in  the  (jiven  expression ;  thus,  in  Ex.  2, 
we  have  a  =  — 12. 


226  COLLEGE  ALGEBRA. 

Factor  the  following : 

3.  ar'  +  13a;  +  40.  13.  12x'-4:X-5. 

4.  x'-Ax-GO.  14.  6-ic-2a^. 

5.  a^- 11a; +  18.  15.  Sa^^^igx-S. 

6.  2x2  + 7a; -15.  iq  10x^-233; +  6. 

7.  5a;2  +  36a;  +  7.  17.  l-Sx-a:^ 

8.  8a;2-18a;  +  9.  18.  15  + 26a;  -  24 a;^^ 

9.  39-10a;-a;2^  19.  Gx" -llax-SBa". 

10.  2  +  x-6ar'.  20.    20x^ +  41mx  +  20m\ 

11.  x'  +  4.x  +  l.  21.    12a^-{-7xy-10y-. 

12.  9ar-6a;-4.  22.    21x' -mm7ix  +  21mV. 

23.    Fa,ctoT2x^-3xy-2y^-7x  +  4:y  +  6. 
Placing  the  expression  equal  to  zero,  we  have 

2a^  -  x(^3y  +  7)  =  2?/^  -4y-6. 
Solving  by  the  formula  of  Art.  347, 


^_3y  +  7±V(3y  +  7)2+16y^-32y-48 
X-  - 


^3y  +  7±V25r  +  lQy  +  l 
4 

^Sy  +  7±(5y  +  l) 
4 

8y  +  8        -2v  +  6     o     .  o        — V  +  3 
=    ^^     or  ^  ^    =  2^/  +  2  or  — ^  ^  ■  . 

Therefore,  2x2_33,y  _  2^2  _  7^_l_4y  ^g 
=  2[x-(2y  +  2)]^a:-:::^J 
=  (u;-22/-li)(:i.«  +  y-3). 


THEORY  OF  QUADRATIC  EQUATIONS.    227 

Factor  tlie  following : 

24.  x^  +  xy—<jy"-  +  x  +  13y  —  6. 

25.  af  +  3xy  +  2y^  +  3x  +  4:y-\-2. 

26.  G  —  5y  +  X  ~  6 y-  +  5 xy  —  x^.  '-■  ■ 

27.  2x^  —  xy  —  y'  +  ox-{-oy—2. 

28.  3a- +  4a6  +  Z>- +  5a -Z>- 12. 

29.  X? —  bxy -\-Qy^ —  5xz  +  l-iyz-^r  ^z^. 

356.  Quadratic  expressions  may  also  be  factored  by  the 
artifice  of  completing  tlie  square  (Art.  343),  in  connection 
with  Art.  128. 

1.    Factor  9 ar^— 9a;  — 4. 

By  Art.  343,  the  expression  9  a;"  —  9aj  will  become  a  per- 
fect square  by  the  addition  of  [  -  | ;  thus, 

9a;2  _  9x  -  4  =  9ar  -  9a;  +  (~^'-  -  -  4 


=  ("'^~2  +  l)(^''-|-2)     (^^-^-l^S) 
=  (3.^'  +  l)(3a;-4). 

Note  1.   If  the  x^  term  is  negative,  tlie  entire  expression  sliould  be 
enclosed  in  a  parenthesis  preceded  by  a  —  sign. 

2.    Factor  3-12a;-4.^•2. 

3  -  12a;  -  4ar  =  -  (4a;2  +  12  a;  — 3) 

=  _(4a;2  +  12a;  +  9-9-3) 

=  - [(2  a; +  3)2- 12] 

=  (2a-  +  3  +  Vl2)-(-l)(2.T  +  3-Vl2) 

=  (2V3 -1-3 +  2:i;)  (2V3  -  3  -  2a;). 


228  COLLEGE   ALGEBRA. 

Note  2.   If  the  coefficient  of  x^  is  not  a  perfect  square,  the  expres- 
sion should  be  divided  by  this  coefficient. 

3.    Factor  6 ar'- 19  a; +  10. 

6si^  -  19x  + 10  =  Qfx'-—  +  ^ 


=  3lx--\.2fx-- 


3J      \        2 
=  (3a; -2)  (2a; -5). 

Note  3.  Tlie  coefficient  of  a;-  may  sometimes  be  made  a  perfect 
square  by  dividing  it  by  one  of  its  factors. 

Tlius,  in  factoring  8x-  — 20  a;  — 7,  the  coefficient  of  x^  may  be  made 
a  perfect  square  by  dividing  it  by  2. 


EXAMPLES. 
Factpr  the  following : 

4.  a;2  +  12a;  +  35.  9.  36ar +  24.t;-5. 

5.  4ar'-9a;-a  10.  8 :^r  +  38 a;  +  35. 
6.' 16ar-31a;+15.  11.  42  +  23.^- lOa.-^. 

7.  3a;-' +  7a; -6.  12.  15a;-'- 14a;  +  31 

8.  9ar-12a;+L  13.  25 x-' -  20 a;  -  2. 

Note  4.  In  factoring  expressions  of  the  form  ax"  +bx  +  c,  the 
nietliod  of  Art.  35(3  is  preferable  when  the  coefficient  of  x^  is  a  perfect 
square  ;  if  the  coefficient  of  a;^  ig  not  a  perfect  square,  the  method  (if 
Art.  ;J55  is  shorter. 


THEORY   OF   QUADRATIC   EQUATIONS.         229 

357.   Certain  expressions  of  the  fourth   degree   may  be 
factored  by  completing  the  square. 

1.  Factor  a*  +  a-b'  +  h\ 

By  Art.  125,  the  expression  will  become  a  perfect  square 
by  the  addition  of  a-lr  to  its  second  term  ;  thus, 

a'  +  a~W  +  h^  =  {a*  +  2  arb"  +  b')  -  a-b^ 

=  {a'  +  b-Y-a-b'' 

=  {a-  +  &-  +  ab)  (a^  +  &2  _  ab)    (Art.  128) 

=  {a-  +  ab  +  b-)  (cr  —  ab  +  b-).. 

2.  Factor  9  a;*  -  39  o^  +  25. 

9a;*  -  39.^2  +  25  =  (9x*  -  30ar  +  25)  -  9^^ 
=  (3ar^-5)2-(3a^)2 
=  {Sx"  +  3x  -  5)  {3af  -  Sx  -  5). 

Note.   The  expression  may  also  be  factored  as  follows : 
9x^-39x2  +  25=  (9x4  + 30  x2  +  25)-69x2 
=  (3x2+5)2-(xV69)2  ' 
=  (3x2  +  xV69  +  5)(3x2-xV(39  +  5). 

Several  of  the  expressions  in  the  following  set  may  be  factored  in 
two  different  ways. 

3.  Factor  a;-*  +  1. 

x*  +  l  =  {x'-\-2xF  +  l)-2x' 

=  {a^  +  iy-(xV2y 

=  (af  +  xV2  +  l){x'-xV2  +  l). 

EXAMPLES. 
Factor  the  following  : 

4.  a;*  +  a.-2  +  l.  7.    m'  +  nv'n' +25 n\ 

5.  x*  —  7x-  +  l.  8.    l-13b^  +  4.b\ 

6.  4:a^-8a-b^+b\  9.    x'-12xy  +  Ay\ 


230  COLLEGE   ALGEBRA. 

10.  4a^  +  8a-  +  9.  15.  IGx* -4.9m-x^ +  9m\ 

11.  4.m'  + 7m' +16.  16.  9a;*-G.^-  +  4. 

12.  a*-5aV  +  a;*.  17.  9a''  + 14aW  +  25m^ 

13.  .'c^  +  81.  18.  4-32w2+497i^ 

14.  4a^  +  loa-&-  +  16&^  19.  16x* -Adxh/ +  25y\ 

358.  The  equation  o;^  + 1  =  0  may  be  solved,  as  in  Art. 
351,  by  placing  the  factors  of  the  first  member  (Ex.  3, 
Art.  357)  separately  equal  to  zero,  and  solving  the  result- 
ing equations ;  thus, 

ar  +  xV2  +  1  =  0;  whence,  a;  =  ~ ^^  * V2-1 


2 
-V2±^ 

/-2 

2 

V2±V- 

> 
72 

and     X'  —  xV2  +  1  =  0;  whence,  x  = 

-    EXAMPLES. 
Solve  the  following  equations  : 

1.  .r^  +  16  =  0.  4.    a;*  +  «*  =  0. 

2.  a;^-Ga;^  +  l  =  0.  5.    a;-*  -  8.^-  + 4  =  0. 

3.  x'--x^-hl  =  0.  6.    x^-^  +  l=0. 

DISCUSSION  OF  THE  GENERAL  EQUATION. 

'  359.   The  roots  of  the  equation  axr  +  6.x  +  c  =  0  are  given 


2a 

It  is  evident  that : 

I.  Ifb^  —  4:ac  is  positive,  the  roots  are  both  real  (Art.  318), 
beivr/  rational  or  irrational  according  as  6'  — 4ac  is,  or  is  not, 
a  perfect  square. 


THEORY   OF   QUADRATIC   EQUATIONS.  23:1 

II.    I  fir  —  4,  ac  is  zero,  the  roots  are  equal. 
III.    Iflr  —  4i  ac  is  negative,  the  roots  are  both  imaginary. 

"'    360.    The  roots  of  the  equation  ocr  +px  =  q  are 

^  -p+-Vir  +  4q      ^^^       ^  -p--Vp^  +  iq^ 

2  2 

We  will  now  discuss  these  values  for  all  possible  real 
values  of  p  and  q. 

I.  Suppose  q  positive. 

Since  p^  is  essentially  positive  (Art.  109),  the  expression 
under  the  radical  sign  is  positive  and  greater  than  p"^. 

Therefore  the  radical  is  numerically  greater  than  p. 

Hence  r^  is  pjositive,  and  r^  is  negative. 

If  p  is  positive,  rg  is  numerically  greater  than  r-^;  that  is, 
the  negative  root  is  numerically  the  greater. 

If  J)  is  zero,  the  roots  are  numerically  equal. 

If  p  is  negative,  ?"i  is  numerically  greater  than  r, ;  that  is, 
the  positive  root  is  numerically  the  greater. 

II.  Suppose  q~0. 

The  expression  under  the  radical  sign  is  now  equal  to  p'^. 
Therefore  the  radical  is  numerically  equal  to  p. 
If  p)  is  positive,  r^  is  zero,  and  n  is  negative. 
If  p  is  negative,  r^  is  positive,  and  r.2  is  zero. 

III.  Stcppose  q  negative,  and  4g  numerically  Kp"^. 

The  expression  under  the  radical  sign  is  now  positive  and 
less  than  p-. 

Therefore  the  radical  is  numerically  less  than  p. 
If  p  is  positive,  both  roots  are  negative. 
If  p  is  negative,  both  roots  are  positive. 

IV.  Suppose  q  negative,  and  4  q  nunierically  equal  to  p-. 
The  expression  under  the  radical  sign  is  now  equal  to  zero. 
Hence  y-j  is  equal  to  r^.     (Compare  Art.  359,  II.) 


232  COLLEGE  ALGEBRA. 

If  p  is  positive,  both  roots  are  negative. 
If  p  is  negative,  both  roots  are  positive. 

V.    Suppose  q  negative,  and  4,q  numerically  >jp^. 

The  expression  under  the  radical  sign  is  now  negative. 

Hence  both  roots  are  imaginary.   (Compare  Art.  359,  III.) 

The  roots  are  both  rational  or  botli  irrational  according  as 
J)-  +  4  g  is  or  is  not  a  perfect  square. 

EXAMPLES. 

1.  Determine  by  inspection  the  nature  of  the  roots  of  the 
equation  2.x--  —  5a;  —  18  =  0. 

The  equation  may  be  written  o? ^  =  9. 

Since  q  is  positive  and  p  negative,  the  roots  are  one 
positive  and  the  other  negative ;  and  the  positive  root  is 
numerically  the  greater. 

In  this  case,  p-  -(-  4  g  =  —  -f  36  =  — ^ ;  a  perfect  square. 
4  4 

Hence  the  roots  are  both  rational. 

Determine  by  inspection  the  nature  of  the  roots  of  the 
following : 

2.  .r-'  + 2.^-15  =  0.  8.  4ar^-7  =  0. 

3.  a;2-10.T  =  -25.  9.  9 x-^  +  30 .'k  =  -  25. 

4.  3a.-2-4a;  =  0.  10.  9ar  +  8  =  24ic. 

5.  a.-  +  5a;  +  3  =  0.  11.  2x-  +  x  =  0. 

6.  3ar'-5a;  +  4  =  0.  12.  11  -  27.x- -  18.x- =  0. 

7.  6x2 -7:k- 10  =  0.  13^  4x2  + 13a;  + 11  =  0. 

361.  It  sometimes  happens  that  on  solving,  by  the  ordi- 
nary process  of  clearing  of  fractions,  an  equation  involving 
the  unknown  quantity  in  the  denominator  of  a  fraction, 
certain  values  of  the  unknown  quantity  are  obtained  which 
will  not  satisfy  the  given  equation. 


THEORY  O^  QUADRATIC  EQUATIONS.    233 

Thus,  let  it  be  required  to  solve  the  equation 

^ox-2_2x±l      J2_^^  (1) 

Multiplying  each  term  by  x^  —  4,  we  have 

(3a.-  -  2)  (re  +  2)  -  (2a;  +  1)  {x  -  2)  +  12  =  0. 
3ar'  +  4x  -  4  -  2ar=  +  3a;  +  2  +  12  =  0. 

ar'  +  7a;  +  10  =  0.  (2) 


^Tru  -  "^  ±  ^-^9  -  4U 

Whence,  x  =  — 


2 
-7±3 


=  -2  or 


The  value  —  2  does  not  satisfy  the  given  equation ;  for, 

2i±l  and  -12_ 
x-\-2  ar  —  4 


with  this  value  of   o:,  the  fractions  and  -^; are 


infinite  (Art.  212). 

The  equation  may  be  solved  in  such  a  way  as  to  obtain 
only  one  root,  as  follows  : 

It  is  evident  from  (2)  that  the  sum  of  the  fractions  in 
the  first  member  of  (1)  is 

a^  +  7a;  +  lQ^  (a;4-2)(a;  +  5)  _cf  +  5. 
a^_4'         (a;  +  2)(a;-2)      a;-2* 

tc  +  5 
Then  (1)  may  be  written  =  0. 

Clearing  of  fractions,  a;  +  5  =  0. 

Whence,  a;  =  —  5. 

It  follows  from  the  above  that  every  value  of  x  obtained 
by  solving  an  equation  which  involves  x  in  the  denominator 
of  a  fraction,  should  be  verified  in  order  to  make  sure  that 
it  satisfies  the  given  equation. 


234  COLLEGE    ALGEBRA. 

XXI.   PROBLEMS. 

INVOLVING  QUADRATIC  EQUATIONS. 

362.  1 .  A  man  sold  a  watch  for  .$  21,  and  lost  as  much 
per  cent  as  the  watch  cost  him.  Kequired  the  cost  of  the 
watch. 

Let  X  =  the  cost  in  dollars. 

Then,  x  =  the  loss  per  cent, 

and  X  •  -^,  or  -^  —  the  loss  in  dollars. 

100        100 

By  the  conditions,     -^  =  .r  —  21. 
^  100 

Solving,  X  =  70  or  30. 

That  is,  the  cost  was  either  $70  or  §  30  ;  for  either  of  these  answers 
satisfies  the  conditions  of  the  problem. 

2.  A  farmer  bought  some  sheep  for  ^72.  If  he  had 
bought  6  more  for  the  same  money,  they  woidd  have  cost 
him  %  1  apiece  less.     How  many  did  he  buy  ? 

Let  X  =  the  number  bought. 

Then,  —  =  the  price  paid  for  one, 

X 

and  '"^    =  the  price  if  there  had  been  0  more. 

x  +  G 

72        7'^ 
By  the  conditions,      —  =  — - — V  1. 
^  X      x  +  6 

Solving,  X  =  18  or  -  24. 

Only  the  positive  value  of  x  is  admissible,  for  the  negative  value 
does  not  satisfy  the  conditions  of  the  problem. 
Therefore,  the  number  of  sheep  was  18. 

Note  1.  In  solving  problems  which  involve  quadratics,  there  will 
usually  be  two  values  of  the  unknown  (juantity  ;  and  those  values 
only  should  be  retained  as  answers  which  satisfy  the  conditions  of  the 
problem. 


PROBLEMS.  23.f 

Note  2.   If  we  should  modify  the  given  problem  so  that  it  shal 
read: 

"A  farmer  .bought  some  sheep  for  $72.    If  he  had  bought  G  feiver 
for  the  same  money,  tliey  would  have  cost  him  $  1  apiece  more.     How 
many  did  he  buy  ?  "    " 
we  should  find  the  answer  24.     (Compare  Art.  207.) 


PROBLEMS. 

3.  Find  two  numbers  whose  difEerence  is  11,  and  whose 
sum  multiplied  by  the  greater  is  513. 

4.  Find  three  consecutive  numbers  whose  sum  is  equal 
to  the  product  of  the  lirst  two. 

6.  Divide  20  into  two  parts  such  that  one  is  the  square 
of  the  other. 

6.  Find  two  numbers  whose  sum  is  13,  and  the  sum  of 
whose* cubes  is  637. 

7.  Find  four  consecutive  numbers  such  that,  if  the  first 
two  are  taken  as  the  digits  of  a  number,  that  number  is  the 
product  of  the  other  two. 

8.  A  merchant  bought  a  quantity  of  flour  for  f  96.  If 
he  had  bought  8  barrels  more  for  the  same  money,  he  would" 
nave  paid  f  2  less  per  barrel.  How  many  barrels  did  he 
buy,  and  at  what  price  ? 

9.  A  merchant  sold  a  quantity  of  wheat  for  -f  39,  and 
gained  as  much  per  cent  as  the  Avheat  cost  him.  What  was 
the  cost  of  the  wheat  ? 

19.  If  the  product  of  three  consecutive  numbers  is  divided 
by  each  of  them  in  turn,  the  sum  of  the  thrCe  quotients  is 
74.     What  are  the  numbers  ? 

11.  A  crew  can  row  5i  miles  down  stream  and  back  again 
in  2  hours  and  23  minutes ;  if  the  rate  of  the  stream  is  3i 
miles  an  hour,  find  the  rate  of  the  crew  in  still  water. 


236  COLLEGE   ALGEBRA. 

12.  A  man  travels  9  miles  by  train.  He  returns  by  a 
train  which  runs  9  miles  an  hour  faster  than  the  first,  and 
accomplishes  the  entire  journey  in  35  minutes.  Required 
the  rates  of  the  trains. 

13.  At  what  price  per  dozen  are  eggs  selling  when,  if  the 
price  were  raised  three-pence  per  dozen,  one  would  receive 
four  less  for  a  shilling  ? 

14.  A  merchant  sold  goods  for  $  16,  and  lost  as  much  per 
cent  as  the   goods  cost  him.      Required   the   cost  of  the 


15.  A  certain  farm  is  a  rectangle,  whose  length  is  twice 
its  breadth.  If  its  length  should  be  increased  by  20  rods, 
and  its  breadth  by  24  rods,  its  area  would  be  doubled.  Of 
how  many  acres  does  the  farm  consist  ? 

16.  A  man  travelled  by  coach  6  miles,  and  returned  on 
foot  at  a  rate  5  miles  an  hour  less  than  that  of  the  coach. 
He  was  50  minutes  longer  in  returning  than  going.  What 
was  the  rate  of  the  coach  ? 

17.  A  square  court-yard  has  a  gravel-walk  around  it. 
The  side  of  the  court  lacks  one  yard  of  being  six  times  the 
width  of  the  walk,  and  the  number  of  square  yards  in  the 
walk  exceeds  the  number  of  yards  in  the  perimeter  of 
the  court  by  340.  Find  the  area  of  the  court,  and  the 
width  of  the  walk. 

18.  The  circumference  of  the  hind-wheel  of  a  carriage  is 
greater  by  4  feet  than  that  of  the  fore- wheel.  In  travelling 
1200  yards,  the  fore-wheel  makes  75  revolutions  more  than 
the  hind-wheel.     Find  the  circumference  of  each  wheel. 

19.  A  cistern  can  be  filled  by  two  pipes  running  together 
in  2  hours  and  55  minutes.  The  larger  pipe  by  itself  will 
fill  it  sooner  than  the  smaller  by  2  hours.  What  time  will 
each  pipe  separately  take  to  fill  it  ? 


TROBLEMS.  237 

20.  The  telegraph  poles  along  a  certain  railway  are  at 
etxual  intervals.  If  there  were  one  more  in  each  mile,  the 
interval  between  the  poles  would  be  decreased  by  8|  feet. 
Find  the  number  of  poles  in  a  mile. 

21.  A  and  B  gained  in  trade  $1800.  A's  money  was  in 
the  firm  12  months,  and  he  received  in  principal  and  gain 
$  2600.  B's  money,  which  was  $  3000,  was  in  the  firm  IG 
months.     How  much  money  did  A  put  into  the  firm  ? 

22.  The  sum  of  f  100  was  divided  among  a  certain  num- 
ber of  persons.  If  each  person  had  received  $  4.50  less,  he 
would  have  received  as  many  dollars  as  there  were  persons. 
Required  the  number  of  persons. 

23.  My  income  is  f  1000.  After  deducting  a  percentage 
for  income  tax,  and  then  a  percentage,  less  by  one  than 
that  of  the  income  tax,  from  the  remainder,  the  income 
is  reduced  to  f  912.  Find  the  rate  per  cent  of  the  income 
tax. 

24.  If  f  2000  amounts  to  $  2163.20,  when  put  at  com- 
povmd  interest  for  two  years,  the  interest  being  compounded 
annually,  what  is  the  rate  per  cent  j)er  annum  ? 

25.  A  man  travelled  102  miles.  If  he  had  gone  3  miles 
more  an  hour,  he  would  have  performed  the  journey  in  5| 
hours  less  time.     How  many  miles  an  hour  did  he  go  ? 

26.  A  man  has  two  square  lots  of  unequal  size,  together 
containing  15,025  square  feet.  If  the  lots  were  contiguous, 
it  would  require  530  feet  of  fence  to  embrace  them  in  a 
single  enclosiire  of  six  sides.     Find  the  area  of  each  lot. 

27.  A  man  has  a  cask  full  of  wine,  containing  72  gallons. 
He  draws  a  certain  number  of  gallons,  and  then  fills  the 
cask  up  with  water.  He  then  draws  out  the  same  number 
of  gallons  as  before,  and  finds  that  there  are  50  g'allons  of 
pure  wine  remaining  in  the  cask.  How  many  gallons  did 
he  draw  each  time  ? 


238  COLLEGE   ALGEBRA. 

28.  A  set  out  from  C  towards  D  at  the  rate  of  3  miles  an 
hour.  After  he  had  gone  28  miles,  B  set  out  from  D  towards 
C,  and  went  every  hour  -^  of  the  entire  distance ;  and  after 
he  had  travelled  as  many  hours  as  he  went  miles  in  an  hour, 
he  met  A.     Eequired  the  distance  from  C  to  D. 

29.  Find  a  number  such  that  the  sum  of  its  cube,  twice 
its  square,  and  the  number  itself,  is  twenty  times  the  next 
higher  number. 

30.  A  courier  travels  from  P  to  Q  in  14  hours.  Another 
courier  starts  at  the  same  time  from  a  place  10  miles  the 
other  side  of  P,  and  arrives  at  Q  at  the  same  time  as  the 
first  courier.  The  second  courier  finds  that  he  takes  half 
an  hour  less  than  the  first  to  accomplish  20  miles.  Find 
the  distance  from  P  to  Q. 

31.  A  and  B  start  at  the  same  time,  from  two  places  180 
miles  apart,  to  meet  each  other.  A  travels  6  miles  a  day 
more  than  B ;  and  the  numbej  of  miles  travelled  each  day 
by  B  was  equal  to  twice  the  number  of  days  which  elapsed 
before  he  met  A.  How  many  miles  did  each  travel  in  one 
day? 

32.  A  man  bought  a  number  of  $20  shares,  when  they 
were  at  a  certain  rate  per  cent  discount,  for  $1500;  and 
afterwards,  when  they  were  at  the  same  rate  per  cent  pre- 
mium, sold  them  all  but  60  for  f  1000.  How  many  shares 
did  he  buy,  and  hoAV  much  did  he  give  apiece  ? 

33.  The  first  digit  of  a  certain  number  is  the  square  of 
the  second.  The  number  exceeds  that  formed  by  reversing 
its  digits  by  twice  the  amount  by  which  the  number  next 
greater  than  the  given  number  exceeds  that  formed  by 
reversing  its  digits.     Eequired  the  number. 


EQUATIONS   SOLVED   LIKE   QUADRATICS.        239 


XXII.    EQUATIONS  SOLVED  LIKE 
QUADRATICS. 

EQUATIONS   IN   THE   QUADRATIC   FORM. 

363.  An  equation  is   said  to  be   in  the  quadratic  form 
when  it  is  in  the  form 

as?"-  +  &.x"  =  c, 

where  n  is  any  rational  number  (Art.  269). 

For  example,  x^  —  Qo?=  16, 

and  x-3  +  x'  '^  =  72, 

are  equations  in  the  quadratic  form. 

364.  Equations   in  the  quadratic  form   may  be  readily 
solved  by  the  rules  for  quadratics. 

1.  Solve  the  equation  x'^  —  6.^"  =  16. 
Completing  the  square  by  the  rule  of  Art.  343, 

a;6_6a;''  +  9  =  16  +  9=25. 
Extracting  the  square  root, 

a;''  -  3  =  ±  5. 

or'-  =  3  ±  5  =  8  or  -  2. 

Extracting  the  cube  root, 

»  =  2  or  -  ^2: 

Note.   There  are  also  four  imaginary  roots  which  may  be  obtained 
by  the  method  of  Art.  351. 

2.  Solve  the  equation  2x  +  3V;«  =  27. 

Since  ^x  is  the  same  as  x^i,  this  is  in  the  quadratic  form. 


240  COLLEGE   ALGEBRA. 

Multiplying  by  8-,  and  adding  3-  or  9  to  both,  members, 

16x  +  2Wx  +  9  =  21G  +  9  =  225. 

Extracting  the  square  root, 

4  V^  +  3  =  ±  15. 

Wx  =  -  3  ±  15  =  12  or  -  18. 

■y/x  =  3  or 

2 

cj  .  n  81 

Squaring,  ic  =  9  or  — 

•  3.    Solve  the  equation  16  x~  ^  -  22a;~^  =  3. 
Completing  the  square  by  the  rule  of  Art.  343, 


(f)  = 


10.-._22.-*  +  (^l=3  +  f  =  ^'. 


.   -I      11      ^13 
4a;  5-  — =  ±— . 
4  4 

.    -3       11       13  1         ^ 

Ax  ^  =  —  ±  —  =  —  -  or  b. 
4        4  2 


Extracting  the  cube  root. 


S  9. 


1  f3\i 
or     - 

2  V2; 


Raising  to  the  fourth  power, 

_i       1 

X  ^  =  —  or 
10 


Inverting  both  members,    cc  =  IC  or  (  ^ 


Note.  In  solving  equations  of  the  form  x^  =  a,  first  extract  the 
root  corresponding  to  the  numerator  of  the  fractional  exponent,  and 
afterwards  raise  to  the  power  corresponding  to  the  denominator. 
Particular  attention  must  be  paid  to  the  algebraic  signs  ;  see  Arts.  109 
and  2;3G. 


EQUATIONS   SOLVED   LIKE   QUADRATICS.       241 

EXAMPLES. 

Solve  the  following  equations  : 

4.  x'-2rjx'  =  -lU.  12.    4a; -15  =  17 Vx 

5.  8x«  +  37.r^  =  216.  13.    a.- +  a;*  =  702. 

2  4 

6.  .r^  -  97  x--"  +  1296  =  0.  14.  2;c'' +  3a;"  -  5G  =  0. 

7.  12a;--  +  a;-i  =  35.  15.  3x^-94x^  =  64. 

8.  69-20.T-3-x-''=0.  16.  3x' -f-26  =  -16.x~^.     V 

9.  x-^-21x--  =  -10S.  17.  2x-^+61x"^-96  =  0. 
y_10.  32.x^  +  ^  =  -33.  18.  6x-^-5x~ 3  =-1184. 

11.    .^•3- 3x^  =  88.  19.    Sx"^  -  15- 2x^  =  0.    V 

365.   An  equation  may  sometimes  be  solved  with  refer 
ence  to  an  expression,  by  regarding  it  as  a  single  quantity. 

1.    Solve  the  equation  (.t  —  5)^  —  3(.x  —  5)  ^  =  40. 
Completing  the  square  by  the  rule  of  Art.  343,  we  have 

(._5).-3(.-5)i  +  (|J=40  +  |=f. 

Extracting  the  square  root, 

(x-5)^  =  ~±l^  =  8  or  -5. 

2       2 

Extracting  the  cube  root, 

(x-5)'  =  2  or  -a/5. 
Squaring,  a;  — 5  =  4  or  -^25. 

Whence,  a;  =  9  or  5  +  ^^^25. 


242  COLLEGE   ALGEBRA. 

Certain  equations  of  the  fourth  degree  may  be  solved  by 
the  rules  for  quadratics. 

2.    Solve  the  equation  x*+  12ay^  +  34ar—  12a;  —  35  =  0. 

The  equation  may  be  written 

(x'  + 12  x^  +  36  x^)  -  2  .^2  -  12  a;  =  35. 

That  is,  (x-  +  6xy--2  {x-  ■j-6x)  =  35. 

Completing  the  square, 

{x^  +  6xy-  2(x"  +  6x)  +  l=  36. 

Extracting  the  square  root, 

(x^  +  6x)-l  =  ±6. 

a^ -{-6x  =  7  or  —  5. 

Completing  the  square,      x--\-  6x'  +  9  =  16  or  4. 

Extracting  the  square  root,        .r  +  3  =  ±  4  or   ±  2. 

^Vlience,  a;  =  -  3  ±  4  or  -  3  ±  2 

=  1,   ~  7,   —  1,  or   —  5. 

Note  1.  In  solving  equations  like  the  above,  the  first  step  is  to 
complete  the  square  with  reference  to  the  a;*  and  x^  terms.  By  Art. 
343,  the  third  term  of  the  square  is  the  square  of  the  quotient  ohtaincd 
by  dividing  the  x^  term  by  twice  the  square  root  of  the  x*  term. 


3,    Solve  the  equation  x^  —  6.r  +  5^xr  —  6x-\-  20  =  46. 
Addine:  20  to  both  members, 


(a^  _  6a;+  20)  +  5 Var  -  6a;  +  20  =  66. 
Completing  the  square, 


289 


(x"-Gx+  20)  +5Var'-6.x-+20-h=r  =  66  +  ^  =    ^ 

4  4        4 

Extracting  the  square  root, 

Va;2-6a;  +  20  +  ^^  =  ±  ^. 


Vx--—  6  a;  +  20  =  6  or  —  11. 
(juaring,  a;-  — 6a;  +  20  :^  36  or  121 


EQUATIONS   SOLVED  LIKE  QUADRATICS.       243 

Completing  tlie  square, 

x2_6ic  +  9  =  25  or  110. 
Extracting  the  square  root, 

a;-3=±5  or   ± VllO. 
Whence,  x  =  8,   -  2,  or  3  ±  VIlO. 

Note  2.  In  solving  equations  of  the  above  form,  add  sucli  a 
quantity  to  both  meUibers  tliat  the  expression  without  tlie  radical  in 
the  first  member  may  be  the  same  as  that  within,  or  some  multiple 
of  it. 

4.    Solve  the  equation  2a^  +  5cc  —  2  xy/x^  +  5a;  —  3  =  12. 
The  equation  may  be  written 


x^  +  Bx  -2x^%^  +  5x  -  3  +  a.-2  =  12. 
Subtracting  3  from  both  members, 


(ar  +  5a;  —  3)  —  2a;  Va;-  +  5a;  —  3  +  a;^  =  9. 
Extracting  the  square  root, 

Va;^  +  5a;  — 3  —  a;  =  ±  3. 


Va;2  +  5a;-3  =  a;±3. 

a;2  -f-  5»  —  3  =  ar  ±  Ct  +  9. 

Therefore, 

-a;  or  11a;  =  12. 

Whence, 

.  =  -12orl|. 

5.    Solve  the  equation  —^ 1 '-  =  — 

XT  —  X      X-  —  3      2 

a."^  —  3 
Representing  — by  y,  the  equation  becomes 


y    2 

or. 

2f  +  2  =  5y. 

Solving, 

y  =  lov2. 

244  COLLEGE   ALGEBRA. 

That  is,  ■  4^  =  J  or  2. 

x-  —  X      2 

Taking  the  first  value,     2ar  —  6  =  ay^  —  x. 

Or,  a^  +  a;  =  6. 

Solving,  x=2  or  —3. 

Taking  the  second  value,   ar  —  3  =  2  a^  —  2  cc. 

Or,  -ar^4-2a;  =  3. 

Solving,  a;  =  1  ±  V—  2. 

EXAMPLES. 
Solve  the  following  equations  : 

--     6.    (x2  -  2 a;)2  -  18 (ar^- 2  a;)  =  -45. 
7.    a:''  +  8ar''-10.T2-104x  +  105  =  0. 
—  8.    2a;2_,.i^V2a:2+l  =  12. 


3a;  +  6V2ar^-3a;  +  2  =  14. 


10.    (3 x~  +  x-iy-26{3x^-^x-iy'  =  27, 


11.  ar' +  7  +  V^+T  =  20. 

12.  (2a.-2  +  3x--l)-  + 2a;- 4-3.1; -3  =  0. 

13.  -^+^+1  =  1 
x'  +  l         X         2 


14.  2ar'-3a;-21  =  2a;Va;'-3a;  +  4. 

15.  Va;  +  10  +  </x  +  10  =  2. 

16.  a;^-10.r''  +  23x-2  +  10a;-24  =  0. 

17.  6/'a;4--Y-35('a;  +  -')  +  50  =  0. 

18.  ^^±1  +  ^  =  -^, 
a; +  3      X-+5  15 


EQUATIONS   SOLVED   LIKE   QUADRATICS.       245 


19.  ^x^-\-2x-i-d  =  x"-j-2a;-{-3. 

20.  (or'  + 16) '  -  3 (ar"'  +  16) ^  +  2  =  0. 

21 .  (ar^  +  &2)  -  =  2  ax"  +  2  a6-a;  -  a-V. 


22.  9a;  +  4  +  2a;V9a;  +  4  =  15a;2^ 

23.  A/3a;-2a;'^-^3ic-2.T2  =  2. 


24.  a;2-|-6V«^-2a;  +  5=ll  +  2a;. 

25.  a;^  +  14 a;"'  +  71  a;^  +  154 a;  +  120  =  0. 

26.  (a;  -  a)-  +  2 Va;(a;  -  a)  =  a^  -  2 aVx. 

2^  a^  +  4a;  +  l      a,-^  +  3a;  +  l  ^  5 

■  x2  +  3a;  +  l      a;2_^4^_^l      2° 

28.  4:(x  -  l)-'3-  _  .5(a;  -  l)"'  +  1  =  0. 


29.  9a;-4ar-5  +  V4a;"'-9a;+ll  =  0. 

30.  a;^-24ar'  +  94a;-  +  600a;-2975  =  0„ 

31.  (2 a; +  5) -^  +  31  (2 a; +  5)- '  =  32. 


32.  3 a:(3  -  x)  =  11  -  4  Va;-  -  3 a.-  +  5. 

33.  (a;-a)*  +  2V^(a.--a)'-36  =  0. 

34.  9  a^^  +  24  af' -  65  ar'- 108  a;  +  140  =  0„ 


35.    2ar-4a;  +  3Va;2-2x  +  6  =  15. 

go     2.T^-3a;-2       3a;-  +  g;-2  ^  13 
3a;2 -f  a;  -  2       2a;2  -  3a;  -  2  ~  6  ' 


37. 


38. 


|l+j5^4^2         ll-Scc  +  x"  ^  5 
\l-3aj  +  a;^'      \l  +  3a;  +  aj^      2 


^l-(^J-^'^ 


2     ^3 
X-  +  1      4* 


246  COLLEGE  ALGEBRA. 

XXIII.     SIMULTANEOUS  EQUATIONS. 

INVOLVING   QUADRATICS. 

366.  Two  equations  of  the  second  degree  (Art.  179)  witb 
two  unknown  quantities  will  generally  produce,  by  elimina- 
tion, an  equation  of  the  fourth  degree  with  one  unknown 
quantity  ;  the  rules  already  given  are,  therefore,  not  suffi- 
cient for  the  solution  of  all  cases  of  simultaneous  equations 
of  the  second  degree  with  two  unknown  quantities. 

Consider,  for  example,  the  equations 

x'  +  y  =  a,  (1) 

and  x-{-y-  =  b.  (2) 

From  (1),  y  =  a  —  a?. 

Substituting  in  (2), 

rc  +  (a  —  x-y  =  b, 

or,  X  +  a^  —  2  ao:r  -\-  x^  =  b, 

which  is  an  equation  of  the  fourth  degree. 

In  several  cases,  however,  the  solution  of  simultaneous 
equations  with  two  unknown  quantities  may  be  effected  by 
means  of  the  rules  for  quadratics. 

Note.    On  the  use  of  the  double  signs  ±  and  q:. 

If  two  or  more  double  signs  are  used  in  a  single  equation,  it  will  be 
understood  that  the  equation  can  be  read  in  two  ways  ;  first,  reading 
all  the  upper  signs  together ;  second,  reading  all  the  lower  signs 
together. 

Thus,  the  equation  a±h  =  ±c  can  be  read  either 

a  +  b  =  c,  OT  a  —  b  =  —  c. 

And  the  equation  a  ±  5  =  q:  c  can  be  read  either 

a  +  b  =  —  c,  i>v  a  —  b  =  c. 

The  same  notation  will  be  used  in  the  case  of  two  or  more  equa- 
tions, each  involving  double  signs. 


S^x-+   42/2  =7G. 

(1) 

(3/-llx-=    4. 

(-0 

9  a;- +  12?/- =  228. 

12r-44./;-=    16. 

53x-^  =  212. 

a;-' -4. 

.r  =  ±  2. 

(3) 

SIMULTANEOUS   EQUATIONS.  247 

Thus,  the  equations  a;  =  ±  2,  ?/  =  ±  3,  can  be  read  either 
a;  =  +  2,  2/= +  3,  or  x=— 2,  ?/  =  — 3. 

And  tlie  equations  x  =  ±  2,  ?/  ;=  =p  3,  can  be  read  either 
a;  =  +  2,  2/  =  —  3,  or  x  =  —  2,  y  =  +  3. 

367.    Case  I.    When  each  equation  is  in  the  form 
aa?  +  hy-  =  c. 


1.    Solve  the  equations 

Multiplying  (1)  by  3, 
Multiplying  (2)  by  4, 
Subtractin.o- 


Whence, 

Substituting  from  (3)  in  (1),  12  +  4  ?/-  =  76. 

2-f  =  16. 
Whence,  y  =  ±  4. 

Therefore,  x  =  2,  ?/  =  ±  4  ;  or,  x  =  —  2,  y  =  ±  4. 

Note.  In  tliis  case  there  are  four  possible  sets  of  values  of  x  and  y 
whicli  satisfy  the  given  equations  : 

1.  x  =  2,  2/=4.  3.  x  =  -2,  y  =  4. 

2.  x  =  2,  y  =  —  'i.  4.  x  =  -2,  y  =  -4:. 

It  -would  be  incorrect  to  leave  the  result  in  the  form  a;  =  ±2, 
y  —  ±4:;  for,  by  Art.  366,  Note,  this  represents  only  the  first  and 
fourth  of  the  above  sets  of  values. 

EXAMPLES; 

Solve  the  following  equations  : 
2     (4:x'+    f-=    61.  4     f    8a.-2-lly2=      8. 

X    x-  +  6r  =  159.  ■    (I2ar  +  13r  =  248. 

.    3_   55a;2-9?/-  =  -121.         ^     j     x-- +  ?/- =  o(«- +  ?>-). 

l7y-  —  3x-=      105.  '    iix- —  y-  =  ria{r>a  —  ih). 


248  COLLEGE   ALCiEBUA. 

Q)x-y       G5       a;  +  2/ 

((a;  +  6t/)^-(5a;  +  32/)(32/-a;)  =  162. 
*   l(3a;-4?/)2-(6a;-2?/)-  =  -195. 

368.  Case  II.  When  one  equation  is  of  the  second  degree^ 
and  the  other  of  the  first. 

Equations  of  tliis  kind  may  always  be  solved  by  finding 
the  valiie  of  one  of  the  unknown  quantities  in  terms  of  the 
other  from  the  simple  equation,  and  substituting  this  value 
in  the  other  equation. 

1.    Solve  the  eouations  .     \'^^'' ~ ^*^  =  ^^-  (^) 

1    X  +2y  =  7.  (2) 

■    From  (2),  "^  =  ^~^-  (^) 

Substituting  in  (1),    2x^-x (^-^^  =  G f^^^^Y 

Clearing  of  fractions,     4  x-^  —  7  x  +  x-^  =  42  —  6  aj. 
5ar-x  =  42. 

Solving,  '^  x  =  3  or 

5 

7+li 

7  —  3  5 

Substituting  in  (3),  y  = ^  or  — —L- 

Therefore,  x  =  3,  ?/  =  2;   or,  x  =  -—,  y  =  —- 

Note  1.  In  this  case  there  are  only  two  possible  sets  of  values  of  x 
and  y  which  satisfy  the  given  equations  : 

Note  2.  Certain  cxanipk'.s  where  one  equation  is  of  the  third  degree, 
and  the  otiier  of  the  first,  may  be  solved  by  the  metluxl  of  Case  IL 


SIMULTANEOUS   EQUATIONS. 


249 


3. 


EXAMPLES. 

Solve  the  following 

equations : 

f2a;--3/  =  -10 
\3x  +  y  =  l. 

^x  +  y  =  -3. 
\xy  =  -  54. 

9. 

x^y 

^x-y=      1. 
U^  +  2/^  =  113. 

10     ix+y=2a. 

■    lx'  +  y'  =  2(a'  +  b'). 

f  ar  +  xy  —  y- = — 

19. 

il. 

(2a;  -2/  =-1. 

\x-y  =  -7. 
(  x^  -f  =  -  117. 

12. 

(x'  +  3a;2/-r  =  23. 
{x  +  2y  =  7. 

'Ix  -y  =-      3. 

(a)^  + 2/3  =  217. 
la;  +?/  =      7. 

13.  < 

[x     y^lO 
y^x       3 

3x-2y  =  -12. 

f.T-2/  =  l. 

14. 

<27r-3xy=15a-10a''. 
l3x+2y  =12a-13. 

(  a;y  =  a^  +  a. 

369.  Case  III.  When  the  given  equations  are  symmetrical 
(Art.  74)  with  respect  to  x  and  y,  and  one  equation  is  of  the 
second  degree,  and  the  other  of  the  second  or  first. 

Equations  of  this  kind  may  always  be  solved  by  combin- 
ing them  in  such  a  way  as  to  obtain  the  values  oi  x  +  y  and 
x  —  y. 

\x  +  y  =  2.  (1) 

1       xy  =  -15.  (2) 

Squaring  (1),  x' +  2  xy -{- y- —  4t.  (3) 

Multiplying  (2)  by  4,  Axy         =-60.  (4) 

Subtracting,  x^  —  2xy  -\-y-  =  64. 

Extracting  the  square  root,        x  —  y  =  ±  8.  (5) 


1.    Solve  the  equations 


250  COLLEGE   ALGEBRA. 

Adding  (1)  and  (5),  2  a;  =  2  ±  8  =  10  or  -  6. 

Whence,  x  =  5  or  —  3. 

Subtracting  (5)  from  (1),  2y  =  2  q:  8  =  -  6  or  10. 
Whence,  y  =  —  3  ov  5. 

Therefore,  cc  =  5,  y  =  —  3;  or,  a;  =  —  3,  y  =  5. 

Note  1.  In  subtracting  ±8  from  2,  we  have  2:^:8,  in  accordance 
with  the  notation  explained  in  Art.  3G6,  Note.  In  operations  with 
double  signs,  ±  is  changed  to  ip,  and  qp  to  ±,  whenever  +  would  be 
changed  to  — . 

Note  2.  The  above  equations  may  also  be  solved  by  the  method  of 
Case  II.  ;  but  the  symmetrical  method  is  shorter,  and  more  elegant. 


2.    Solve  the  equations  \ 

lx^  +  xy+f  =  2S. 

(1) 
(2) 

Dividing  (1)  by  (2),                    x-y  =  2. 

(3) 

Squaring  (3) ,                  ar  —  2  xy  +  ?/"  =  4. 

(4) 

Subtracting  (4)  from  (2),             3  a;?/ =  24, 

a;^  =  8. 

(5) 

Adding  (2)  and  (5) ,       a.-'  -f  2  xy  +  y-  =  36. 

Whence,                                        x  +  y  =  ±6. 

(C) 

Adding  (3)  and  (G),      2  a;  =  ±  G  +  2  =  8  or 

-4. 

Whence,                                                 a;  =  4  or 

-2. 

Subtracting  (3)  from  (G), 

2?/  =  ±G-2  =  4or 

-8. 

Whence,                                                 ?/  =  2  or 

-4. 

Therefore,  a;  =  4,  y  =2 ;   or,  a;  =  —  2,  y'~  — 

4. 

Note  3.  The  above  equations  are  not  symmetrical  according  to  the 
definition  of  Art.  74  ;  but  the  method  of  Case  III.  may  often  be  used 
in  cases  where  the  given  equations  are  symmetrical  except  with  respect 
to  the  signs  of  the  terms. 

Note  4.  Certain  examples  in  which  one  equation  is  of  the  third 
degree,  and  the  other  of  the  first  or  second,  may  be  solved  by  the 
method  of  Case  III. 


.    SIMULTANEOUS   EQUATIONS.  251 

«     o,  1        .  .  ( ar -r  ?/- =  50.  (1) 

3.    Solve  the  equations       - 

(         xy  =  ~7.  (2; 

Multiplying  (2)  by  2,  2  a;?/ =  -14  (3) 

Adding  (1)  and  (3),  cc^  +  2,x\j  +  1/  =  36. 

Whence,  x  +  ?/  =  ±  6.  (4) 

Subtracting  (3)  from  (1), 

a? -2  xy  +  y'  =  64. 

Whence,  x  —  y  =  ±?>.  (5] 

Adding  (4)  and  (5),  2a;  =  6  ±  8,  or  -  6  ±  8. 

Whence,  x  =  7,  —  1,  1,  or  —  7. 

Subtracting  (5)  from  (4),  2?/  =  6  ip  8,  or  -  6  rp  8. 

Whence,  ?/  =  —  1,  7,  —  7,  or  1. 

Therefore,  x=±T,  ?/  =  :p  1  ;    or,  x=±l,  ?/  =  ip  7. 

EXAMPLES. 

Solve  the  following  equations  : 

^     <xy  =  4.8.  ^Q     (ar^  +  r  =  -513. 

'    lx  +  y=U.  '    \x  -{-y  =-      9. 

5    j-^-'  +  /=    122.  ^j     ^x^  +  f=    260. 

'U+y=-10.  '   la;  -2/  =-14. 

e.    (ar''-r  =  -65.  ^^^    ( ar' +  y'^  =  504. 

1  ar  +  a-^  +  ?/- =  13.  '   \x"  -  xy  +  y-  =  8-i. 

y^    (.^y  =  -24.  j3     (a;2  +  r=    305. 

(  x-  -  2/  =  11.  '    i  ?/  -  a;  =  -  21. 

8     l^-y  =  6.  j^     fa;^+2/-  =  185. 

U^  +  2/'  =  13.  '   la;?/  =  -88. 

'    (x  -y  =    2.  '   Xx--  xy  +  2/'  =  124. 

*  Divide  the  first  equation  by  the  secdiid. 


252 

16 

\x  +  y  = 

-h\ 

2  a. 

COLLEGE   ALGEBRA. 


18.    1^2/ =  -150. 
lx-y  =  -^l. 

^^     5ar'  +  /==13(a2  +  l).  ^^     (o;^- /=  3a(a  +  l)  +  l. 

Ix  +y  =oa  —  l.  \y  —  x  =  —  l. 

370.  Case  IV.  When  each  equation  is  of  the  second  degree 
and  homogeneous  (Art.  25). 

Note  1.  Certain  equations  whioli  are  of  the  second  degree  and 
homogeneous  may  be  solved  by  the  methods  of  Cases  I.  and  III. 
(See  Ex.  1,  Art.  367,  and  Ex.  3,  Art.  369.)  The  method  of  Case  IV. 
should  be  used  only  when  the  example  cannot  be  solved  by  the  methods 
of  Cases  I.  or  III. 

(  oir  —2xy  =    5. 
1.    Solve  the  equations       ■{    „ 

Putting  y  =  vx  in  the  given  equations,  we  have 

x^  —  2vay^=    5  ;  or,  x-  =  — '- ;  (1) 

1  —  2v 

29 

and  ar  +  v-a^  =  29  ;  or,  a^ 


Equating  the  values  of  xr, 


29 


l-2v       1  +  ^2 
5  +  5^2=29-58^. 


Or, 

5u-  +  58v 

=  24. 

Solving  this 

equation. 

V  ■■ 

4- 

-12. 

Substituting 

these  values 

in  (l^,     .^-': 

B 
1-i 

or        ^ 

°'l  +  24 

=  25  or 

1 
5' 

Whence, 

X 

=  ±  5  or  ±  — . 

V5 


SIMULTANEOUS   EQUATIONS.  253 


Substituting  the  values  of  v  and  x  in  the  equation  y  =  vx, 

12 


7/  =  J  (  ±  5)  or  -  12f  ±  —  )  =  ±  2  or  T  ^^ 


1  12     — 

Therefore,  a;  =  ±  5,  y  =  ±  2;  or,  a;  =  ±  -Vo,  ?/=  ip  — V5, 

5  5 

Note  2.    In  finding  y  from  the  equation  y  =  ■ya;,  care  must  be  taktii 
to  multiply  each  pair  of  values  of  x  by  the  correspondincj  value  of  v. 


EXAMPLES. 

Solve  the  following  equations  : 

1    a;2  +  2r  =  18.  "  t  2 a;- -  ?/- =  23. 

g     j  2a;-+ a'?/  =  15.  „  (oy^j^^xy—    ?/-  =  — 7. 

'    (    ar-  ?/-=    8.  '  1x2  + 3a.'?/ -2?/- =  -4. 

^     (  a.-2  +  a;?/ —  ?/2  =  —  11.  [x-2  — a;?/  — 16?/2=    4. 

'   la^  +  7/2  =  13.  *  \x^-{.xy-    8^  =  22. 


5. 


a;2  +  32/2  =  28.  g     (3a;2+    a;?/  +  /  =  47. 

a.-2  +  a;?/  +  2?/==16.  '    Ua;^  -  3a;?/ -2/^  =  _  39. 

jQ     f  53a.-2-128a;2/  +  642/2=5. 
113x2-    31x?/  + 162/2  =  21 


MISCELLANEOUS   EXAMPLES. 

371.  No  general  rules  can  be  given  for  the  solution  of 
examples  which  do  not  come  under  the  cases  just  consid- 
ered. Various  artifices  are  employed,  familiarity  with  which 
can  only  be  gained  by  experience. 

1.    Solve  the  equations  ■{    „  o        /.  /o\ 

(x-y-xy-=    6.  (^) 

Multiplying  (2)  by  3,  ?,x"y  -  ?,xy-  =  18.  (3) 


254  COLLEGE   ALGEBRA. 

Subtracting  (3)  from  (1), 

x^  —  3  u?y  +  3  xy-  —  y^  =  1. 
Extracting  the  cube  root,        x  —  y  =  l.  (4) 

Dividing  (2)  by  (4),  xy  =  &.  ^5) 

Solving  equations  (4)  and  (5)  by  the  method  of  Case  III., 
we  find  x  =  3,  y  =  2]  ov,  x  =  —  2,  y  =  —  3. 

(  x^  +  y^  =  9  xy. 

2.  Solve  the  equations 

^  Ix  +y  =6. 

Putting  x  =  u  -\-  V  and  y  =  u  —  v,  we  have 

(u  +  vy  +  (u - vy  =  9{u-hv) (u-vy  (i) 

and  (lo  +  v)  +{u  —  v)   =6.  (2) 

Eeducing  (1),  2?r  +  6?«v'  =  9(?t-  — -?;-).  (3) 

Reducing  (2),  2u  =  G,  or  ?t  =  3. 

Substituting  the  value  of  u  in  (3), 

Whence,  v^  =  1,  or  v  =  ±  1. 

Therefore,  a;  =  r(  +  'y  =  3±l  =  4or2; 

and  y  =  u  —  V  =  3  ^:1  =  2  or  4. 

Note.  The  artifice  of  substituting  u  +  v  and  ii  —  v  for  x  and  y  is 
advantageous  in  any  case  wliere  the  given  equations  are  symmetrical 
(Art.  74)  with  respect  to  x  and  y.     See  also  Ex.  4. 

3.  Solve  the  equations 


i^x'  +  y'-  +  2x  +  2y  =  23. 

(1) 

X                              xy  =  Q. 

(2) 

;)by2,                 2xy  =  12. 

(3) 

Adding  (1)  and  (3), 

x'  +  2 xy  -^y'  +  2x+2y  =  35, 
or,  {x  +  yy  +  2{x  +  ,v)  =  .35. 


SIMULTANEOUS  EQUATIONS.  255 

CoiTi^^leting  the  square, 

{x-i-yy  +  2{x  +  y)  +  l  =  36. 
Whence,  (a-  +  y)  +  1  =  ±  6  ; 

or,  '  cc  +  ?/  =  5  or  —  7.  (4) 

Squaring  (4),  x'  +  2  xy  -\- y"- =  25  or  49. 

Multiplymg  (2)  by  4,  4:xy  =  24. 

Subtracting,  x^  —  2xy  +  if=    1  or  25. 

Whence,  x-y=  ±1oy  ±5.  (5) 

Adding  (4)  and  (5),  2a;  =  5  ±  1,  or  -7  ±  5. 

Whence,  a;  =  3,  2,  -  1,  or  -  6. 

Subtracting  (5)  from  (4),  2y  =  5=fl,or  -7^5. 

Whence,  y  =  2,  3,  —  6,  or  —  1. 

( x'^  -\-  ?/  =  97 
4.    Solve  the  equations       •]  ' 

(x  +y  =  -  1. 

Putting  x  =  u  +  v  and  y  =  u  —  v,  we  have 

{jj,  +  yy^(^u-vy  =  97,  (1) 

and  («  +  ^^)  +(m  — v)   =  —  1.  (2) 

Eeducing  (1),    2w*  +  12itV  + 2'y*  =  97.  (3) 

Reducing  (2),  2u  =  -  1,  or  u  =  -  -  . 

Substituting  in  (3),   ^Jf-3v-  +  2v^  =  97. 

o 

Solving  this  equation,  if-  —  i^  q-^-  _  ^. 

4  4 

Whence,  ^  ^  ±  5  ^^.  ^  V^31^ 

2  2 

Therefore,     x=:u-\-v  =  -- ±-,  ov  -\±^-'''^^ 
=  2,  or  —  3,  or 


l±  V^^3T 


256  COLLEGE   ALCxEBRA. 

Also,  y  =  u-v  =  -^T-,ov  -^t'^~^^ 

2      2  2  2 


=  -  3,  or  2,  or 
5.    Solve  the  equations 


x{x-\-y  +  z)  =  a\ 

(1) 

' 

y(x  +  y  +  z)  =  b\ 

(2) 

, 

z{x-i-y  +  z)  =  c^. 

(3) 

Dividing  (1)  by  (2), 

y      b^                a^ 

Dividing  (1)  by  (3), 

Substituting  in  (1), 

<^+!^-+S)="' 

or,                                    x'{a'  +  h''  +  c^)  =  a\ 

AVhence, 

.-±      «'     . 

Va-  +  &'  +  c^ 

Therefore, 

v-'''~± '- , 

a-           Va-+62  +  c^ 

and 

z-^--±           ''           . 

a-          Va^  +  ?>^  +  c^ 

EXAMPLES. 

Solve  the  following  equations  : 

'  .T  _^  ?/  _  29 
6.  J  2/      X      10* 

g      (  x''-2xy=16. 
■   1  2a;y4-2/-  =  -3. 

I3x-2y=:4.. 

'   \  xy  =  (}.  '   \  xy  +  A.y-  =30. 


11. 

4.12. 


Ol3. 


14. 
15. 

23. 
i24. 

25. 
26. 


1+1=11 

X      y 


SIMULTANEOUS   EQUATIONS, 
16 


257 


—  =  18. 

xy 

X?  -\- y^  z=  9  —  X. 
X?  —  y-  =  Q>. 

orY  +  28  .T?/- 480  =  0. 
2x  +  y  =  ll. 


XT      y- 


65. 


i  -  -  =  11. 

aj       y 

.T*  +  2/"  =  IT. 
X  -y  =    3. 

a:2  4.42/2  +  3a;=22. 
2x7/ +  3?/  + 9=    0. 


17. 


18. 


19. 


21. 


\x  +  2y  = 
1  xy  +  2/*  = 

3a  +  &. 
=  2a(a  +  6) 

91. 

[Wr- 

7. 

xy{x—y)=2b{a-—b-). 

xy  +  a;/  =  12.         y 
X   +  a;2/3  =  18.     'A 

X*  +  ar'?/-  +  ?/*  =  133. 
x^  +  xy  +  y^  =  7. 

.x-^  —  xy-{-y'  =  19. 
2.^■2-2/-  =  -17. 


„2     ,  a^  +  x?/  +  ?/2  =  7a2-13a&  +  76-. 
a~  —  .x?/  +  y^  =  3  a-  —    3  ab  +  3  6^. 


x^  +  f  =  33. 
X  +2/  =    3. 

9a^—     xy  —  y  =  51. 

3  a;  —  5  »?/  +  ?/-  =  81. 

x-y=  218. 

Va;  —  Vy  =  2. 


27. 


2/      ;» 


30. 


3-'/  + 2/  = 

=  1. 

a;-y +  r= 

=  5. 

x-  -  xy  = 

=  27y. 

xy  -  y-  = 

:      3X. 

x  —  y      X 
x-\-y      X 

+  2/_ 

-y 

3 

2 

2x-y  = 

7. 

2ab 
X = 

y 

=  a. 

y 


2ab 


X  +y  =1. 

*  Divide  the  first  equation  by  tlie  second. 


=  6. 


258  COLLEGE   ALGEBRA. 


31.   \<l=^{^-^)-  36.   J-'^"-2/  =  2. 


32. 


(  .r  =  7/(3  -  2/).  i  r  +  -ixix'  -  4)  =  13. 

2a;  +  32/  +  i^  =  -6.  37.    j  ^G^"^  +  2/0  =  1' .'^-V- 
?/  (.  .^  +  2/  =  3. 

a;  +  52/  +  i^  =  -13.  38.    f  a7/-(-'^-!/)  =  l- 

y  1    ..;Y+(.^_y)2^13. 

33.   \  ^'(/  -  1 )  =  «2/-  39_   I  xy  +  .X-  -  2/  =  -  o. 

o.y  (.^•^  —  1)  =  .r.  '1  ar?/  —  xy-  =  —  84. 


35. 


Va;'  +  22  2/-  4-  i«-  =  22.  L  Vafy  +  V/'x  =  40. 

2_J^_2_g  ^j     I  2a;2  +  22/'  =  5xy. 

a;-      xy      y-        '  i  ^^  -hy  =  33. 

A  _  1  + 1  =  44  42.   p^'-  2"^^  -  ''""^  ''^*'''^'- 

xy      x^      y^  \  y- -{- 3  xy -\- X- =  11  x-y\ 

^3     (  .r'  +    re-?/  +    xy-  +  f  =  -  41- 
1  (t-  +  2  .or>  +  2  .x-7/2  +  /•  =  -  21. 

44     I  i>j'  -  2/'  +  X  4-  y  =  -  2. 

(a;  +  ?/)(2.x-  +  32/-l)  =  3. 


Vx-^  +  ?/2  4- x  —  y  =  12. 
45. 


V(a;  -  ?/)-(.x-'  +  /)  =  35. 

r  Va^  — Va;-2/  =  ll. 
46.  ^      

l^  ^x'y  —  y'-x  =  60. 

(  X--2 xy  +  3xz  =  —lG.  \  {y-\- 1>)  (2  +  c)  =  a^. 

47.  -]  2 .T  -  3 2/  =  7.  49.  \   {z  +  c)  (x  +  « )  =  6-. 

[  3.r  4-  r>z  -  -  14.  [  (X  4a)  (.'/  4  ^)  =  c-. 

r    a;^4r  +  z^=14.  (  (y  +  z)  {x  +  y +  z)=2. 

48.  -j  2:«- 3.7  +  2  =  11.  50.  }   {z  +a:){x  +  y  +  z)=3. 

x  +  2y-z=:-G.  [{x  +  y){x  +  y  +  z)=i. 


SIMULTANEOUS  EQUATIONS.  259 


PROBLEMS. 

"  372.  1.  The  sum  of  the  squares  of  two  numbers  is  106, 
and  the  difference  of  their  squares  is  ^  the  square  of  their 
difference.     Find  the  numbers. 

2.  The  difference  of  the  squares  of  two  numbers  is  55, 
and  the  product  of  their  squares  is  576.    Find  the  numbers. 

3.  If  the  length  of  a  rectangular  field  were  increased  by 
2  rods,  and  its  breadth  by  3  rods,  its  area  would  be  108 
square  rods  ;  and  if  its  length  were  diminished  by  2  rods, 
and  its  breadth  by  3  rods,  its  area  would  be  24  square  rods. 
Find  the  length  and  breadth  of  the  iield.     . 

cJ  4.  The  sum  of  the  cubes  of  two  numbers  is  407,  and  the 
sum  of  their  squares  exceeds  their  product  by  37.  Required 
the  numbers. 

6.  If  the  product  of  two  numbers  is  multiplied  by  their 
sum,  the  result  is  520 ;  and  the  sum  of  the  cubes  of  the 
numbers  is  6.37.    Find  the  numbers. 

6.  A  man  bought  6  ducks  and  2  turkeys  for  $  15.  He 
bought  four  more  ducks  for  $14  than  turkeys  for  $9. 
What  was  the  price  of  each  ? 

7.  Find  a  number  of  two  figures,  such  that,  if  its  digits 
are  inverted,  the  sum  of  the  number  thus  formed  and  the 
original  number  is  33,  and  their  product  252. 

n  8.  The  sum  of  two  numbers  exceeds  the  product  of  their 
square  roots  by  7 ;  and  if  the  product  of  the  numbers  is 
added  to  the  sum  of  their  squares,  the  result  is  133.  Find 
the  numbers.. 

9.  The  sum  of  the  terms  of  a  fraction  is  17.  If  the 
numerator  is  increased  by  5,  and  the  denominator  dimin- 
ished by  5,  the  product  of  the  resulting  fraction  and  the 
original  fraction  is  •^.     Required  the  fraction. 


260  COLLEGE   ALGEBRA. 

10.  A  rectangular  garden  is  surrounded  by  a  walk  7  feet 
wide ;  the  area  of  the  garden  is  15,000  square  feet,  and  of 
the  walk  3696  square  feet.  Find  the  length  and  breadth  of 
the  garden. 

11.  A  rectangular  field  contains  an  acre.  If  its  length 
were  increased  by  4  rods,  and  its  breadth  by  3  rods,  its  area 
would  be  increased  by  100  square  rods.  Find  the  length 
and  breadth  of  the  field. 

12.  A  man  rows  down  stream  12  miles  in  4  hours  less 
time  than  it  takes  him  to  return.  Should  he  row  at  twice 
his  usual  rate,  his  rate  down  stream  would  be  10  miles  an 
hour.  Find  his  rate  in  still  water,  and  the  rate  of  the 
stream. 

13.  A  distributes' $  180  equally  amongst  a  certain  num- 
ber of  persons.  B  distributes  the  same  amount  amongst  a 
number  of  people  less  by  40,  and  gives  to  each  person  $  6 
more  than  A  does.  What  amount  does  A  give  to  each 
person  ? 

14.  A,  B,  and  C  together  can  do  a  piece  of  work  in  one 
hour.  B  does  twice  as  much  work  as  A  in  a  given  time ; 
and  B  alone  requires  one  hour  more  than  C  alone  to  per- 
form the  work.  In  what  time  could  each  alone  perform 
the  work  ? 


15.  Two  couriers,  A  and  B,  start  at  the  same  time  from 
wo  towns,  P  and  Q,  respectively,  and  travel  towards  each 

other.  «k  When  they  meet,  it  is  found  that  A  has  travelled 
72  miles  more  than  B ;  also,  that  A  will  arrive  at  Q  in  9 
days,  and  that  B  will  arrive  at  P  in  16  days.  Required  tlie 
distance  between  P  and  Q,  and  the  rates  of  the  couriers. 

16.  If  the  product  of  two  numbers  is  added  to  their  sum 
the  result  is  47 ;  and  the  sum  of  their  squares  exceeds  thei 
sura  by  62.     Required  the  numbers. 

Note.    Keprescut  the  numbers  by  x  +  y  and  x  —  ij. 


SIMULTANEOUS   EQUATIONS.  261 

17.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
fourth  powers  is  641.     Find  the  numbers, 

18.  The  difference  of  two  numbers  is  2,  and  the  differ- 
ence of  their  fifth  powers  is  242.     Find  tlie  numbers. 

19.  A  sets  out  to  walk  to  a  town  7  miles  off,  and  20  min- 
utes afterwards  B  starts  to  follow  him.  When  B  has  over- 
taken A,  he  turns  back  and  reaches  the  starting-point  at 
the  same  instant  that  A  reaches  his  destination.  If  B 
walked  at  the  rate  of  4  miles  an  hour,  what  was  A's  rate  ? 

20.  Three  vessels  ply  between  the  same  two  ports.  The 
first  sails  half  a  mile  an  hour  faster  than  the  second,  and 
makes  the  trip  in  li  hours  less  time.  The  second  sails 
three-quarters  of  a  mile  an  hour  faster  than  the  third,  and 
makes  the  trip  in  2|-  hours  less  time.  Eequired  the  dis- 
tance between  the  ports. 

21.  A  and  B  run  a  race  of  four  miles.  A  reaches  the 
half-way  post  five  minutes  before  B ;  he  then  diminishes 
his  speed  3  miles  an  hour,  while  B  increases  his  speed  4 
miles  an  hour,  and  beats  A  by  seven  minutes.  Eequired 
the  rates  of  A  and  B  at  first. 

22.  A  cistern  can  be  filled  by  three  pipes,  A,  B,  and  C, 
when  opened  together,  in  6  hours.  If  A  filled  at  the  same 
rate  as  B,  it  would  take  8^  hours  for  A,  B,  and  C  to  fill  the 
cistern;  and  the  sum  of  the  times  required  by  A  and  C 
alone  to  fill  the  cistern  is  double  the  time  required  by  B 
alone.  What  time  will  each  pipe  alone  require  to  fill  the 
cistern  ? 


262  COLLEGE   ALGEBRA. 


XXIV.     INDETERMINATE  EQUATIONS  OP 
THE  FIRST  DEGREE. 

373.  It  has  already  been  shown  that  a  single  equation 
containing  two  or  more  unknown  quantities  is  satisfied  by 
an  indefinitely  great  number  of  sets  of  values  of  these 
quantities  (Art.  189) ;  and,  in  general,  that  a  set  of  m  in- 
dependent equations  containing  more  than  m  unknown 
quantities  is  satisfied  by  an  indefinitely  great  number  of 
sets  of  values  of  the  unknown  quantities  involved  in  it 
(Art.  204). 

Such  equations  are  called  indeterminate. 
If,  however,  the  unknown  quantities  are  required  to  sat- 
isfy other  conditions,  the  number  of  solutions  may  be  finite. 

374.  We  shall  consider  in  the  present  chapter  the  solu- 
tion of  indeterminate  equations  of  the  first  degree,  containing 
two  unknown  quantities,  in  which  the  unknown  quantities 
are  restricted  to  positive  integral  values. 

Every  such  equation  can  be  reduced  to  one  of  the  forms 

ax  ±by  =  c,  or  ax  ±by  =  —  c, 

where  a,  b,  and  c  represent  positive  integers  which  have  no 
common  divisor. 

The  equation  ax  -\-by  =  —  c  cannot  be  solved  in  positive 
integers ;  for,  if  x,  y,  a,  and  b  are  positive  integers,  ax  +  by 
must  also  be  a  positive  integer. 

Again,  the  eqxiations  ax  ±by  =  c  and  ax  —  by  =  —  c  can- 
not be  solved  in  positive  integers  if  a  and  b  have  a  common 
divisor. 

For,  if  x  and  y  are  positive  integers,  this  common  divisor 
must  also  be  a  divisor  of  ax  ±  by,  and  consequently  of  c ; 
which  is  contrary  to  the  hypothesis  that  a,  b,  and  c  have  no 
common  divisor. 


INDETERMINATE   EQUATIONS.  263 


SOLUTION    OF    INDETERMINATE    EQUATIONS    IN 
POSITIVE    INTEGERS. 

375.   1.  Solve  7x  -{-  5y  =  118  in  positive  integers. 

Dividing  tlirougli'by  5,  tlie  smaller  of  the  two  coefficients, 
we  ho.ve 

5  5 

or,  ^^ —  =  23  -  a;  -  y, 

o 

Since  by  the  conditions  of  the  problem  x  and  y  must  be 

positive  integers,  it  follows  that  ~ '-  must  be  an  integer. 

Let  this  integer  be  represented  by  iJ. 

Then,  ^'^"'^  =  p,  or  2  x  -  3  =  5p.  (1 ) 

o 

Dividing  (1)  by  2,  a;  -  1  -  ^  =  2^. +1 ; 

or,  x-1-  273  =  Vj±1, 

^  2 

Since  x  and  p  are  integers,  cc  —  1  —  2^9  is  also  an  integer ; 

and  therefore  -^      —  must  be  an  intes:er. 

2 

Let  this  integer  be  represented  by  q. 

Then,  ^^^  =  q,ovp  =  2q-  1. 

Substituting  in  (1),   2x  —  o  =  y)q  —  5. 

Whence,  x  =  5q  —  l.  (2) 

Substituting  this  value  in  the  given  equation, 

35g-7  +  5?/  =  118. 
Whence,  y  —  25  —  7q.  (3) 


204  COLLEGE    ALGEBRA. 

Equations  (2)  and  (3)  form  Avhat  is  called  the  geneial 
solution  ill  integers  of  the  given  eq^uation. 

By  giving  to  q  the  values  zero,  or  any  positive  or  negative 
integer,  we  shall  obtain  sets  of  integral  values  of  x  and  y 
which  satisfy  the  given  equation. 

Now  if  q  is  zero,  or  any  negative  integer,  x  will  be  nega- 
tive ;  and  if  q  is  any  positive  integer  greater  than  3,  y  will 
be  negative. 

Hence  the  only  jwsitive  integral  values  of  x  and  y  which 
satisfy  the  given  equation  are  those. arising  from  the  values 
1,  2,  3  of  q. 

That  is,  a;  =  4,  2/  =  18  ;  x  =  9,y  =  ll;  and  a;  =  14,  ?/  =  4. 

2.    Solve  8  a;  —  13?/  =  100  in  positive  integers. 

Dividing  through  by  8,  the  coefficient  of  smaller  absolute 
value,  we  have 

x  —  y ^  =  12  +  -: 

^       8  8' 

-IO      5y  +  4 
or,  a;  —  w  —  12  =    ^    — . 

y  8 

Then  '^^  J^     must  be  an  integer. 
8 

Multiplying  by  5,     '  ^  ^  '"    must  also  be  an  integer. 

That  is,  3?/  +  "  +  2  +  -  must  be  an  integer  ;  and  therefore 
8  8 

"         must  be  an  integer. 
8  ^      ,  . 

Let  this  integer  be  represented  by  p. 

Then,  -^         =  p,  or  y  =  8|)  —  4. 

8 

Substituting  in  the  given  equation, 

8x-  104i)  4-  52  =  100,  or  x  =  13p  +  6. 

In  this  case  ;:>  may  be  any  positive  integer. 


INDETERMINATE   EQUATIONS.  265 

Thus,  if  p  =  1,  x  =  19  and  ?/  =  4 ;  if  p  =  2,  a;  =  32  and 

y  =  12;  etc. 

The  number  of  solutions  is  therefore  indefinitely  great. 

Note.  The  artifice  of  multiplying  '^^T"  by  5  saves  considerable 
work  in  the  above  example. 

The  rule  in  any  case  is  to  multiply  the  numerator  of  the  fraction  by 
such  a  number  that  the  coefficient  of  the  unknown  qliantity  shall 
exceed  some  multiple  of 'the  denominator  by  unity. 

If  this  had  not  been  done,  the  last  part  of  the  solution  of  Ex.  2  would 
'  have  stood  as  follows  : 

Let  ^]L±A  =  p^  or  5y  +  i  =  8p.  (1) 

8 

Dividing  by  5,  2/  +  ^  =  p  +  ^• 

5  5 

yhcn     ^~     must  be  an  integer. 
5 

Let  ^£-~  =  q,  OT  3p-i  =  6q.    '  (2) 

Dividing  by  3,        p  -  \ -1  =  q +  "11. 

Then  ^^         must  be  an  integer. 

Let  2g|J:  =  r,  or  2g  +  l  =  3r.       ^  (3) 

1  r 

Dividing  by  2,  g  -f  .-  =  r  +  — 

Then  ^~     must  be  an  integer. 

Let  tZL^^s,  orr  =  2s+l. 

Substituting  in  (3) ,    2  g  +  1  =  6  s  +  3,       or  g  =  3  s  +  1. 

Substituting  in  (2),    3p  —  4  =  15  s  +  6,     or  p  =  5  s  +  3. 

Substituting  in  (1),    5  ?/  +  4  =  40  s  +  24,  or  ?/  =  8  s  +  4.      . 

Substituting  in  the  given  equation, 

8a; -104s- 52  =  100,  or  a;  =  13s +19. 

The  values  of  x  and  y  differ  in  form  from  those  obtained  above  ;  but 
it  is  to  be  observed  th^t  13s +19  and  8s +  4,  for  the  values  0,  1,  2, 
etc.,  of  s,  give  rise  to  the  same  series  of  positive  integers  as  1.3p  +  (? 
and  825  —  4  for  the  values  1,  2,  3,  etc.,  ol  p. 


266  COLLEGE   ALGEBRA. 

We  will  now  show  how  to  solve  in  positive  integers  two 
equations  involving  three  unknown  quantities. 

3.  In  how  many  ways  can  the  sum  of  $14.40  be  paid 
with  dollars,  half-dollars,  and  dimes,  the  number  of  dimes 
being  equal  to  the  number  of  dollars  and  half-dollar^- 
together  ? 

Let  X  =  the  number  of  dollars, 

y  =  the  number  of  half-dollars, 
and  2  =  the  number  of  dimes. 

Then  by  the  conditions, 

10a;  +  5?/  +  ^  =  144, 
and  x  +  y  =  z-  (1) 

Adding,  llx -{-(J7j+z  =  lU  +  z, 

or,  llx  +  6y=lU.  (2) 

Dividing  by  6,      x  +  ^^  +  y  =  24. 
0 

Then  —  must  be  an  integer ;  or,  x  must  be  a  multiple  of  6 

6 
Let  X  =  Gj?,  where  p  is  an  integer. 

Substituting  in  (2), 

QGp  +  Gy  =  144,  or  ?/  =  24  -  lip. 

Substituting  in  (1),  z  =  6p  -f  24  -  lip  =  24  -  5^^ 

The  only  positive  integral  solutions  are  when  p  =  l  or  2. 

Therefore  the  number  of  ways  is  two ;  either  6  dollars, 
i:>  half-dollars,  and  19  dimes ;  or  12  dollars,  2  half-dollars, 
and  14  dimes. 

EXAMPLES. 

Solve  the  following  in  ])ositive  integers  : 

4.  2.T-f3?/=21.  -6.    7;«  +  :5S?/  =  211. 

5.  7x  +  4y  =  S0.  7.    .".1 ..; -h '>// -  1222. 


INDETERMIXATE   EQUATIONS.  2G7 

8.  24a; +  7?/ =  422.  10.    A6x+lly  =  1117. 

9.  8x  +  G7y=WS.  11.   8x  +  19?/=700. 

Solve  the  following  in  least  positive  integers : 

12.  4.x- 3y  =  5.  15.    21a;  -  8^/  =  -  25. 

13.  5x-Ty  =  ll.  16.    13a;- 30i/ =  61. 

14.  19a; -42/ =  128.  17.    17a;  -  58 7/  =  -  79. 

Solve  the  following  in  positive  integers : 

jg     f  2a;  4- 3?/ -52  =  -8.        ^g     f  3a;-22/ -  32  =  -  65. 
1 5a;-    7/ +  42=    21.  '   \8x  +  5y  +  2z=    177. 

20.  In  how  many  different  ways  can  the  sum  of  $  3.90  be 
paid  with  fifty  and  twenty  cent  pieces  ? 

21.  In  how  many  different  ways  can  the  sum  of  19  s.  6d. 
be  paid  with  florins  worth  2  s.  each,  and  half-crowns  worth 
2s.  6d.  each? 

22.  Find  two  fractions  whose  denominators  are  9  and  5, 
respectively,  and  whose  sum  shall  be  equal  to  J^V-. 

23.  In  how  many  different  ways  can  the  sum  of  ^5.10  be 
paid  with  half-dollars,  quarter- dollars,  and  dimes,  so  that 
the  whole  number  of  coins  used  shall  be  20  ? 

24.  A  farmer  purchased  a  certain  number  of  pigs,  sheep, 
and  calves  for  1 160.  The  pigs  cost  $  3  each,  the  sheep  $  4 
each,  and  the  calves  $  7  each ;  and  the  number  of  calves  was 
equal  to  the  number  of  pigs  and  sheep  together.  How  many 
of  each  did  he  buy  ? 

25.  In  how  many  different  ways  can  the  sum  of  £8  2s. 
be  paid  with  half-crowns,  florins,  and  shillings,  so  that  twice 
the  number  of  half-crowns  together  with  five  times  the  num- 
ber of  florins  shall  exceed  three  times  the  number  of  shillings 
by  11? 


X 


268  COLLEGE   ALGEBRA. 


XXV.   RATIO  AND  PROPORTION. 

376.  The  Ratio  of  one  number  to  another  is  the  quotient 
obtained  by  dividing  the  first  number  by  the  second. 

Thus,  the  ratio  of  a  to  5  is  -;  and  it  is  also  expressed 
a:b.  ^ 

377.  A  Proportion  is  an  equality  of  ratios. 

Thus,  if  the  ratio  of  a  to  6  is  equal  to  the  ratio  of  c  to  d, 
they  form  a  proportion,  which  may  be  written  in  either  of 
the  forms : 

a  :  b  =  c  :  d,       =  -,    or    a  :  b  :  :  c  :  d. 
b      d 

378.  The  first  term  of  a  ratio  is  called  the  antecedent,  and 
the  second  term  the  consequent. 

Thus,  in  the  ratio  a  :  b,  a  is  the  antecedent,  and  b  is  the 
consequent. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms  the  means. 

Thus,  in  the  proportion  a:b  =  c:d,  a  and  d  are  the 
extremes,  and  b  and  c  the  means. 

379.  In  a  proportion  in  which  the  means  are  equal,  either 
mean  is  called  a  Mean  Proportional  between  the  first  and 
last  terms,  and  the  last  term  is  called  a  Third  Proportional 
to  the  first  and  second  terms. 

A  Fourth  Proportional  to  three  quantities  is  the  fourth 
term  of  a  proportion  whose  first  three  terms  are  the  three 
quantities  taken  in  their  order. 

Thus,  in  the  proportion  a  :  b  =  b  :  c,  b  is  a  mean  propor- 
tional between  a  and  r,  and  c  is  a  third  proportional  to  a 
and  b. 


RATIO    AND    rKOrORTION.  269 

In  the  proportion  a  :  b  =  c  :  d,  d  is  a  fourth  proportional 
to  a,  b,  and  c. 

380.  A  Continued  Proportion  is  a  series  of  equal  ratios, 
in  which  each  consequent  is  the  same  as  the  following 
antecedent ;  as, 

a  :  b  =  b  :  c  =  c  :  d  =  d  :  e. 


PROPERTIES  OF  PROPORTIONS. 

381.    In  any  proportion  the  prrodiict  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Let  the  proportion  be    a:b  =  c:  d. 


Then  by  Art.  377, 


a  _c. 
b~d 

Clearing  of  fractions,        ad  =  be. 


382.  A  mean  proportional  between  two  quantities  is  equal 
to  the  square  root  of  their  jrroduct. 

Let  the  proportion  be    a  :  b  =  b  :  c. 
Then  by  Art.  381,  b'  =  ac. 

Whence,  b  =  -Vac. 

383.  From  the  equation  ad  =  be,  we  obtain 

be        -,  r      ad 
a  =— ,  and  b  =  —  • 
d  c 

That  is,  in  any  proportion  either  extreme  is  equal  to  the 
product  of  the  means  divided  by  the  other  extreme;  and 
either  mean  is  equal  to  the  product  of  the  extremes  divided 
by  the  other  mean. 

384.  (Converse  of  Art.  381.)  If  the  product  of  two  quan- 
tities is  equal  to  the  product  of  two  others,  one  pair  may  be 
made  the  extremes,  and  the  other  pair  the  means,  of  a  pro- 
portion. 


ad  = 

--be. 

ad 
bd 

=i-'- 

a 
b^ 

c 
d 

a:b  = 

--c:d. 

270  COLLEGE   ALGEBRA. 

Let 

Dividing  by  bd, 
Whence, 

In  like  manner  we  may  prove  that 
a  :  c=  b  :  d, 
c  :  d  =  a  :  b,  etc. 

385.  In  any  proportion  the  terms  are  in  proportion  by 
Alternation;  that  is,  the  first  term  is  to  the  third  as  the  second 
term  is  to  the  fourth. 

Let  a  :  b  =  c  :  d. 

Then  by  Art.  381,  ad  =  be. 

Whence  by  Art.  384,      a:c=b:d. 

386.  In  any  proj)ortion  the  terms  are  in  proportion  by 
Inversion ;  that  is,  the  second  term  is  to  the  first  as  the  fourth 
term  is  to  the  third. 

Let  a:b  =  c:d. 

Then,  ad  =  be. 

Whence,  b  :  a  =  d:  c. 

387.  In  any  proportion  the  terms  are  in  jwoportion  by 
Composition ;  that  is,  the  sum  of  the  first  two  terms  is  to  the 
first  term  as  the  smn  of  the  last  two  terms  is  to  the  third  term. 

Let  a:b  =  c:d. 

Then,  ad  =  be. 

Adding  both  members  to  ac, 

ac  +  ad  =  ae  +  be, 
or,  a(e  +  d)=^  c{a  -\-b). 

Whence  (Art.  384),  a  + b:a  =  e -\- d:  c. 
Similarly  we  may  prove  that 

a  -j-  b  :  b  =  c  +  d  :  d. 


RATIO   AND   PROPORTION.  271 

388.  In  any  proportion  the  terms  are  in  proportion  by 
Division;  tJiat  is,  the  difference  of  the  first  two  terms  is  to  the 
first  term  as  the  difference  of  the  last  two  terms  is  to  the  third 
'erm. 

Let  a  :  b  =  c  :  d. 

Then,  ad  =  be. 

Subtracting  both  members  from  ac, 

ac  —  ad  =  ac  —  be, 
or,  a{c  —  d)  =  c{a  —  b). 

Whence,  a  —  b:a  —  c  —  d:c. 

Similarly,  a  —  b  -.b  =c  —  d  :  d. 

389.  In  any  proportion  the  terms  are  in  proportion  by 
Composition  and  Division ;  that  is,  the  sum  of  the  first  two 
terms  is  to  their  difference  as  the  sum  of  the  last  tioo  terms  is 
to  their  difference. 

Let  a:b  =  c:d. 

Then  by  Art.  387,         a  +  b  ^c  +  d^  ,^. 

a  c 

And  by  Art.  388,  ^~^  =  ^  ~ 'I  (2) 

a  c 

Dividing  (1)  by  (2),   ^_^. 
a  —  b      c  —  d 

Whence,  a  -]-  b :  a  —  b  =  c  -}-  d  :  c  —  d. 

390.  In  a  series  of  equal  ratios,  any  antecedent  is  to  its 
consequent  as  the  sum  of  all  the  antecedents  is  to  the  sum  of 
all  the  consequents. 

Let  a:b  =  c:  d  =  e  :f. 

Then  by  Art.  381,  ad  =  be, 

and  af=  be. 

Also,  ab  =  ba. 

Adding,  a(b  +  d +f)  =  b(a-\- c  +  e). 

Whence  (Art.  384),       a:  b  =  a  +  c  + e:b -\-d-\-f 


272  COLLEGE   ALGEBRA. 

In  like  manner  the  theorem  may  be  proved  for  any  num- 
ber  of  equal  ratios. 

'391,   To  jyrove  that  if 

b~d~f~"'' 
then  each  of  these  equal  ratios  is  equal  to 

fpa"  +  gc"  +  re"  + 
\2^b"+qd'^+rf^+ 

Let  ^  =  '-='-=....=k. 

b      d     f 

Then,  a  =  bk,  c  =  dk,  e=fk,  etc. 

Whence, 

pa"  +  gc"  +  re"  +...  =;)(6A:)"  +  q{dky  +  r(fk)"  + 

=  k\2^b"  +  qd'^  +  rf'^ +■■■)■ 
Therefore,  ^»  ^  j?a"  +  gc"  +  re"  + -. 

i?&"  +  gcZ" +'//"  +  ••. 

Or,  k  =  (Pd''  +  qc^+re-  + 

\2ib''  +  qd"+rf'^  + 

If  2',  g,  r,  etc.,  are  all  equal,  and  n=l,  we  have 
f^_<^_e^_       _a  +  c  +  e+  ••• 
6  ~  d~/ ~  ■"  ~  ftT^zTTT^' 

(Compare  Art.  390.) 

392.   /7i  any  number  of  proportions,  the  2>roducts  of  the  cor- 
respjonding  terms  are  in  p)roportion. 

Let  a:b  =  c:d, 

and  e:f=fj:h. 

Then,  ^*  =  ",and^  =  2. 

b      d  f      h 

Multiplying  these  equals, 

a      e       c  ^a  ae      cq 

-  X  ~  =  -  X  ■',  or  —  =  -^. 
6     /      d      h'        bf      dh 

Wlience,  ae  :  bf=  eg  :  dh. 


RATIO   AND   PROPORTION.  273 

In  like  manner  the  theorem  may  be  proved  for  any  num- 
ber of  proportions. 

393.  In  any  proportion,  if  the  first  two  terms  are  multiplied 
by  any  quantity,  as  also  the  last  tivo,  the  resulting  quantities 
'■joill  be  in  proportion. 

Let  a:b  =  c:d. 

Then, 

Therefore, 

Whence,  ma  :  mb  =  nc  :  nd. 

In  like  manner  we  may  prove  that 
ah      c    d 


Note.    Either  in  or  n  may  be  unity  ;  that  is,  either  couplet  may  be 
multiplied  or  divided  without  multiplying  or  dividing  the  other. 


394.  In  any  proportion,  if  the  first  and  third  terms  are 
midtiplied  by  any  quantity,  as  also  the  second  and  fourth 
terms,  the  resulting  quantities  will  be  in  proportion. 

Let  a:b  =  c:d. 


a 
6" 

c 
~d 

ma 
mb 

nc 
"  nd 

Then, 

a      c 
b~d 

Therefore, 

ma  _  mc 
nb      nd 

Whence, 

ma  :  nb  =  mc  :  % 

In  like  manner  we  may  prove  that 

a    b       c     d 

m'  n~ m'  n 

Note.   Either  m  or  n  may  be  unity. 


274  COLLEGE  'ALGEBRA. 

395.  In  any  proportion,  like  powers  or  like  roots  of  the 
terms  are  in  proportion. 

Let  a  :  b  =  c  :  d. 

Then,  '^  =  '-. 

b     d 

Therefore,  "       ^i!  =  ^. 

Whence,  a" :  6"  =  c"  :  d". 

In  like  manner  we  may  prove  that 

Va  :  'Vb  =  Vc  :  Vd. 

396.  If  three  quantities  are  in  continued  proportion,  the 
first  is  to  the  third  as  the  square  of  the  first  is  to  the  square  of 
the  second. 

Let  a:b  =  b:c. 

Then,  ^^  =  ^. 

'  b      c 

rpt,       p  a      b      a     a         a      a^ 

Therefore,  -  x  -  =  -  X  -,  or  -  =     • 

b      c      b      b         c      b- 

Whence,  •     a:  c  =  a- :  &'. 

397.  If  four  quantities  are  in  continued  jiroportion.,  the 
first  is  to  the  fourth  as  the  cidje  of  the  first  is  to  the  cube 
of  the  second. 

Let  a  :  b  =  b  :  c  =  c:  d. 

a  _b_c 
bed 


Then, 


rpv.       n  a      b      c      a      a      a 

Therefore,  _x-X-  =  7X-X-- 

0      c      d      b      b      b 


Or, 

Whence,  a  :  d  =  a^:  b^. 


a  _  a" 
d~b^ 


Note.    The  ratio  a^ :  b-  is  called  the  duplicate  ratio,  aud  the  ratio 
«' :  h^  the  triplicate  ratio,  of  a  •  h. 


n 


RATIO   AND   PROPORTION.  275 

PROBLEMS, 
398.   1.  Solve  the  equation  2.?;+3:2a;-3  =  a+2&:2&  — a. 
By  Art.  389,  4  .t  :  G  =  4  6  :  2  o. 

Dividing  the  first  and  third  terms  by  4,  and  the  second 
and  fourth  terms  by  2  (Art.  394),  we  have 

X  :  o  =  b  :  a. 

Whence,  x  =  —  (Art.  383). 

2.  If  x:y  =  (x-\-z)' -.(y +  zy',  prove   that   z  is    a   mean 
proportional  between  x  and  y. 

From  the  given  proportion,  by  Art.  381, 

y{x-\-z)-  =  x{y  +  zy. 
Or,  a^y  -\-2  xyz  +  yz-  —  xy--j-  2  xyz  +  xz^. 

Or,  _  ary  —  xy^  =  xz-  —  yz-. 

Dividing  hj  x  —  y,  xy  =  z^. 

Therefore  z  is  a  mean  proportional  between  x  and  y. 

3.  Given  the  equations  j  4x  -  Sy  +  5z  =  0.  (1) 

l3x-5y-4.z  =  0.  (2) 

To  find  the  ratio  of  x  to  y,  and  the  ratio  of  x  to  z. 

Multiplying  (1)  by  i,  16x -12y  +  20z  =  0. 

Multiplying  (2)  by  5,  15a;  -25y-20z  =  0. 

Adding;  31  a;  -  37  ?/  =  0,  or  31  a;  =  37 y 

Whence  by  Art.  384,  '      x:y  =  37:  31. 

Multiplying  (1 )  by  5,    20  a;  - 15  y  +  25  2  =  0. 

Multiplying  (2)  by  3,      9x-16y-12z  =  0. 

Subtracting,  11  a;  +  37  2  =  0, 

or,  37z  =  -llx- 

Whence,  a; :  2;  =  37  :  —  11. 

Note.   The  result  may  be  written  in  the  form  —  =  -■'-  =  ^_. 

37      31      -11 


276  COLLEGE   ALGEBRA. 


4.    Prove  that  if  "  =  -,  then 
b      d 


a^  -  62 :  a2  -  3a&  =  c^  -  d' :  c^  -  3cd. 


Let  —  =  -  =  x:  whence,  a  =  bx. 
b      d 


Then, 


6  V  -  62 


a'- 

-3ab      b-x^-Sb'x     x^-Sx 

t-1 

d'               c'  -  d' 

c"      3c     c2-3cd 
d'      d 

Whence,  or—  b- :  a?—  3ab  =  c^  —  d^ :  c^  —  o cd. 

5.    Find  a  fourth  proportional  to  |,  f,  and  |' 

-  6.    Find  a  third  proportional  to  f  and  f  • 

7.  What  is  the  second  term  of  the  proportion  whose 
first,  third,  and  fourth  terms  are  5|,  44,  and  If  ? 

8.  Find  a  third  proportional  to  a^  —  9  and  3  —  a. 

9.  Find  a  mean  proportional  between  5|  and  IS-j^- 

10.  Find  a  mean  proportional   between    and 

a;  +  2 

Solve  the  following  equations  : 

11.  5x-3a:5a;-|-3a^9a-25:21a-25. 

12.  2cc-5:3a;  +  2  =  x-l:7a;  +  l. 

-  13.    x'-4:x--9=:x'-5x  +  6:x"-\-4:X  +  3. 


14.  x  +  Vl-X':x-Vl-  X-  =a+  V6-  -a-:  a-  ^l)^—d\ 

(  X  +  y:  X  —  y  z=  a  +  b  :  a  —  b. 

15.  J  1 

I  cc2  +  7/2:a2=a2  +  6^i.        ' 


RATIO    AND   PROPORTION.  277 

~  16.  Find  two  numbers  in  the  ratio  of  16  to  9  such  that, 
when  -each  is  diminished  by  8,  they  shall  be  in  the  ratio  of 
12  to  5. 

17.  Divide  36  into  two  parts  such  that  the  greater  dimin- 
ished by  4  shall  be  to  the  less  increased  by  3  as  3  is  to  2. 

18.  Find  two  numbers  such  that,  if  4  is  added  to  each, 
they  will  be  in  the  ratio  of  5  to  3 ;  and  if  11  is  subtracted 
from  each,  they  will  be  in  the  ratio  of  10  to  3. 

19.  There  are  two  numbers  in  the  ratio  of  3  to  4,  such 
that  their  sum  is  to  the  sum  of  their  squares  as  7  is  to 
50.     What  are  the  numbers  ? 

20.  Divide  12  into  two  parts  such  that  their  product 
shall  be  to  the  sum  of  their  squares  as  3  is  to  10. 

21.  Divide  a  into  two  parts  such  that  the  first  increased 
by  b  shall  be  to  the  second  diminished  by  6,  as  a  +  3  6  is  to 
a -3b. 

22.  If  5a  +  4c:9a  +  2c  =  46  +  5c:2&  +  9c,  prove  that 
c  is  a  mean  proportional  between  a  and  b. 

23.  If  (a  +  6  +  c  +  d)  (a  -  6  -  c  +  d)  =  (a  -  6  +  c  -  d) 
(a  -t-  6  —  c  —  cZ),  prove  that  a:b  =  c:d. 

24.  If  ax  — by:  ex  —  dy  =  ay  —  bz:cy  —  dz,  prove  that  y  is 
a  mean  proportional  between  x  and  z. 

25.  If  a-c:b-  d  =  Va-  -f  c^ :  V6'  +  d'',  prove  that 
a  :  b  =  c  :  d. 

'26.  If  8  cows  and  5  oxen  cost  four-fifths  as  much  as  9 
cows  and  7  oxen,  what  is  the  ratio  of  the  price  of  a  cow  to 
that  of  an  ox  ? 

27'.  Given  (a^+ab)x-\-  {b--ab)y  z=  (a--{-b-)x  -  {a^-b-)y  ^ 
find  the  ratio  of  x  to  y. 

-    28.    Given   l^^"    y  +  4.z  =  0. 
(2x  +  5y-3z  =  0. 
Fiud  the  ratio  of  x  to  y,  and  the  ratio  of  x  to  z. 


278  COLLEGE    ALGEBRA. 

29.  Giy^,,c,y-bx^cx-az^hz-cy^ 

c  b  a 

Find  tlie  ratio  of  x  to  y,  and  tlie  ratio  of  x  to  z. 

30.  Divide  $  564  between  A,  B,  and  C,  so  that  A's  sliaie 
may  be  to  B's  in  the  ratio  of  5  to  9,  and  B's  share  to  C'e  in 
the  ratio  of  7  to  10. 

31.  Each  of  two  vessels  contains  a  mixture  of  wine  and 
water,  A  mixture  consisting  of  equal  measures  from  the 
two  vessels,  contains  as  much  wine  as  water ;  another  mix- 
ture consisting  of  four  measures  from  the  first  vessel  and 
one  from  the  second,  is  composed  of  wine  and  water  in  the' 
ratio  of  2  to  3.  Find  the  ratio  of  wine  to  water  in  each 
vessel. 

32.  The  population  of  a  town  increased  2.6  per  cent  from 
1870  to  1880.  The  number  of  males  decreased  3.8  per  cent 
during  the  same  period,  and  the  number  of  females  increased 
10.6  per  cent.  What  was  the  ratio  of  males  to  females  in 
1870? 

33.  The  sum  of  four  quantities  in  proportion  is  30.  The 
third  term  exceeds  the  sum  of  the  first  and  second  by  2, 
and  the  sum  of  the  fourth  and  second  terms  exceeds  the 
first  term  by  6.     What  are  the  quantities  ? 

34. 


If 

a      c 
b~d' 

prove  that 

(a 

.)«'  + 

2ab:3ab- 

Ab- 

=  c- 

-{-2cd:ocd- 

■4(Z-. 

(b 

.)  ap- 

ab +  b'-:  — 

-W 

=  c^ 

/.3 

-  cd  +  d-  :  — 

-d" 

35.  It -  =  -  =  -,  prove  that 

b      d     f  ' 

(a.)  a'  +  r  +  e' :  b'  +  d'  +f  =  ace  :  bdf. 

{b.)  (a-  +  c^  +  e')  (b'-  +  (Z-  +f-)  =  (ab  +  cd  +  e/y. 

36.  If  a,  b,  c,  and  d  are  in  continued  proportion,  prove 
that  2 a  +  od:  3 a  -  4 (Z  =  2 (t^  +  3 b'^ :  3 tv'  —  4 UK 


VARIATION.  279 


XXVI.     VARIATION. 

399.  One  quantity  is  said  to  vary  directly  as  another  when 
the  ratio  of  any  two  values  of  the  first  is  equal  to  the  ratio 
of  the  corresponding  values  of  the  second. 

Note.  It  is  customary  to  omit  the  word  "directly,"  and  say  simply 
that  one  quantity  varies  as  auotlier. 

400.  Suppose,  for  example,  that  a  workman  receives  a 
fixed  sum  per  day. 

The  amount  which  he  receives  for  m  days  will  be  to  the 
amount  which  he  receives  for  n  days  as  m  is  to  7t ;  that  is,, 
the  ratio  of  any  two  amounts  received  is  equal  to  the  ratio 
of  the  corresponding  numbers  of  days  worked. 

Hence  the  amount  which  the  workman  receives  varies  as 
the  number  of  days  during  which  he  works. 

401.  One  quantity  is  said  to  vciry  inversely  as  another 
Avhen  the  first  varies  directly  as  the  reciprocal  of  the  second. 

Thus,  the  time  in  which  a  railway  train  will  traverse  a 
fixed  route  varies  inversely  as  the  speed;  that  is,  if  the 
speed  is  doubled,  the  train  will  traverse  its  roitte  in  one-half 
the  time. 

402.  One  quantity  is  said  to  vary  as  two  others  jointly 
when  it  varies  directly  as  their  product. 

Thus,  the  wages  of  a  workman  varies  jointly  as  the 
amount  which  he  receives  per  day,  and  the  number  of  days 
during  which  he  works. 

403.  One  quantity  is  said  to  vary  directly  as  a  second 
and  inversely  as  a  third,  when  it  varies  jointly  as  the  sec- 
ond and  the  reciprocal  of  the  third. 

Thus,  in  physics,  the  attraction  of  a  body  varies  directly 
as  the  quantity  of  matter,  and  inversely  as  the  square  of 
the  distance. 


280  COLLEGE  ALGEBRA. 

404.  The  symbol  oc  is  used  to  express  variation ;  thus, 
ace  6  is  read  "a  varies  as  &." 

405.  If  xcr,  y,  then  x  is  equal  to  y  multiplied  by  a  constant 
quantity. 

Let  a;'  and  y'  denote  a  fixed  pair  of  corresponding  values 
of  X  and  y,  and  x  and  y  any  other  pair. 
•    Then  by  the  definition  of  Art.  399, 

X      y  x' 

—  =  ^,  or  x  —  -y. 
x'      y'  ^      y' 

Denoting  the  constant  ratio  -  by  m,  we  have 
X  =  my. 

406.  It  follows  from  Arts.  401,  402, 403,  and  405  that : 

1 .  If  X  vanes  inversely  as  y,  x  =  — 

2.  If  X  varies  jointly  as  y  and  z,  x  =  myz. 

3.  Ifx  varies  directly  ns  y  and  inversely  as  z,  x=—- 

Note.  The  converse  of  each  of  the  statements  of  Arts.  405  and  406 
is  also  true ;  that  is,  if  x  is  equal  to  y  multiplied  by  a  constant  quan- 
tity, then  XX  y;  and  so  on. 

407.  To  2irove  that  ifxccy,  and  yccz,  then  xccz. 

By  Art.  405,  ii  xccy,  then  x  =  my ;  (1) 

and  if  yc^z,  y  =  nz. 

Substituting  in  (1),  x  =  mnz. 

Whence  by  Art.  406,  Note,  xooz. 

408.  To  jyi'ove  that  if  xckz,  and  yccz,  then  x±yccz,  and 
■\/xyccz. 

By  Art.  405,  x  =  7nz,  and  y  =  nz. 

Therefore,  x±y=  mz  ±nz  =  (m  ±n)z, 

and,  -y/xy  =  Vmz  •  yiz  =  z^mn. 

Whence,  x±y  (jzz,  and  Vx?/  oc  z. 


VARIATION.  281 

489.    To  prove  that  if  xccy,  and  z  x  ?i,  then  xz  x  yu. 
We  Lave,  x  =  my,  and  z  =  nu. 

Therefore,  xz  =  mnyu. 

Whence,  xz  cc  yri. 

410.  To  prove  that  ifxccy,  then  a?"  cc  y". 

We  have,  a;  =  my ;  or,  a;"  =  m^'y*. 

Whence,  x"  cc  ?/". 

411.  To  prove  that,  if  xocy  when  z  is  constant,  and  xccz 
when  y  is  constant,  then  xccyz  ivhen  both  y  and  z  vary. 

Let  y'  and  z'  be  the  values  of  y  and  z,  respectively,  when 
X  has  the  value  x'. 

Let  y  be  changed  from  y'  to  y",  z  remaining  constantly- 
equal  to  z',  and  let  x  be  changed  in  consequence  from  aj'to  X. 

Then  by  Art.  399,  ^  =  llL.  (1) 

X    y" 

Now  let  z  be  changed  from  z'  to  z",  y  remaining  constantly 
equal  to  y",  and  let  x  be  changed  in  consequence  from  X  to 
x". 

Multiplying  (1)  by  (2),    ^=^-  (3) 

Now  if  both  changes  are  made,  that  is,  y  from  y'  to  y"  and 
2  from  z'  to  2",  a;  is  changed  from  x'  to  x",  and  ?/2;  is  changed 
from  y'z'  to  i/"z;". 

Then  by  (3),.  the  ratio  of  any  two'values  of  x  is  equal  to 
the  ratio  of  the  corresponding  values  of  yz. 

Therefore  by  Art.  399,  x  cc  yz. 

In  like  manner  it  may  be  proved  that  if  there  are  any 
number  of  quantities  a;,  ?/,  z,  u,  etc.,  such  that  xccy  when  2, 
u,  etc.,  are  constant,  a;  cc  z  when  y,  u,  etc.,  are  constant,  etc., 


282  COLLEGE  ALGEBRA. 

then  if  all  tlie  quantities  y,  z,  u,  etc.,  vary,  x  varies  as  their 
product. 

The  following  is  an  illustration  of  the  above  theorem : 
It  is  known,  by  Geometry,  that  the  area  of  a  triangle 

varies  as  the  base  when  tlie  altitude  is  constant,  and  as  the 

altitude  when  the  base  is  constant. 

Hence,  when  both  base  and  altitude  vary,  the  area  varies 

as  their  product. 

'412.  Problems  in  variation  are  readily  solved  by  con- 
verting the  variation  into  an  equation  by  aid  of  Arts.  405 
or  40G. 

EXAMPLES. 

413.  1.  If  a;  varies  inversely  as  y,  and  is  equal  to  9  when 
y  =  S,  what  is  the  value  of  'x  when  ?/  =  18  ? 

If  X  varies  inversely  as  y,  we  have  by  Art.  406^, 

m 
x  =  —• 

y 

Putting  £c  =  9  and  y  =  S,  we  obtain 


Whence, 


Hence,  if  2/= 18,  we  have 


'    , 


18 


2.  Given  that  the  area  of  a  triangle  varies  jointly  as  its 
base  and  altitude,  what" will  be  the  base  of  a  triangle  whose 
altitude  is  12,  equivalent  to  the  sum  of  two  triangles  whose 
bases  are  10  and  6,  and  altitudes  3  and  9,  respectively  ? 

Let  B,  H,  and  A  denote  the  base,  altitude,  and  area, 
respectively,  of  any  triangle,  and  B'  the  base  of  the  required 
triangle. 


VARIATION.  283 

Since  A  varies  jointly  as  B  and  H,  we  have 
A  =  mBH  (Xrt.iOG). 

Then  the  area  of  the  first  triangle  is  vixlOxo,  or  30  m, 
and  the  area  of  the  second  is  mxGxO,  or  Bim;  and  hence 
the  areaof  the  required  triangle  is  30  m  +  54  m,  or  84  m. 

But  the  area  of  the  required  triangle  is  also  m  X  B'  X  12. 

Therefore,  12  mB'  =  84  m. 

Whence,  B'  =  7. 

3.  If  oj  varies  inversely  as  y,  and  is  equal  to  4  when 
y  =  2,  what  is  the  value  of  y  when  .^•  =  |-  ? 

4.  If  y  cc  2-,  and  is  equal  to  15  when  z  —  o,  what  is  the 
value  of  y  in  terms  of  z'^  ? 

5.  If  2  varies  jointly  as  x  and  _?/,  and  is  equal  to  90  when 
a;  =  3  and  ?/  =  6,  what  is  the  value  of  z  when  x  =  2  and  ?/=7  ? 

6.  If  a;  varies  directly  as  y  and  inversely  as  z,  and  is 
equal  to  4  when  ?/=2  and  z=o,  what  is  the  value  of  x  when 
?/  =  35  and  2;  =  15  ? 

7.  If  2a;  —  3  cc  3?/  +  7,  and  x=o  Avhen  y=l,  what  is  the 
value  of  X  when  y  —  —  l? 

8.  If  ar'  cc  ?/-,  and  .r  =  G  when  2/  =  3,  what  is  the  value 
of  y  when  x  —  2? 

9.  The  distance  fallen  by  a  body  from  a  position  of  rest 
varies  as  the  square  of  the  time  during  which  it  falls.  If  a 
body  falls  257i  feet  in  four  seconds,  how  far  will  it  fall  in 
seven  seconds  ? 

10.  Two  quantities  vary  directly  and  inversely  as  x,  re- 
spectively. If  their  sum  is  equal  to  7  when  x  =  2,  and  to 
— 13  when  x  =  ~3,  what  are  the  quantities  ? 

11.  The  area  of  a  circle  varies 'as  the  square  of  its  diam- 
eter. If  the  area  of  a  circle  whose  diameter  is  2^  is  lO/^, 
what  will  be  the  diameter  of  a  circle  whose  area  is  34||  ? 


284  COLLEGE   ALGEBRA. 

12.  Given  that  y  is  equal  to  the  sum  of  two  quantities 
which  vary  directly  as  a?  and  inversely  as  x,  respectively. 
If  ?/  =  —  1-  when  x=i,  and  y  =  ^'^-  when  x  =  —  2,  what  is 
the  value  of  y  when  x  =  —  |-  ? 

13.  Given  that  y  is  equal  to  the  sum  of  three  quantities, 
the  first  of  which  is  constant,  an'd  the  second  and  third  vary 
as  X  and  a.-^,  respectively.  If  ?/  =  —  19  when  x  =  2,  ?/  =  4 
w^hen  x  =  l,  and  ?/  =  2  when  x  =  —  l,  what  is  the  expression 
for  y  in  terms  of  a;  ? 

14.  If  the  volume  of  a  pyramid  varies  jointly  as  its  base 
and  altitude,  what  will  be  the  altitude  of  a  pyramid  whose 
base  is  12,  equivalent  to  the  sum  of  two  pyramids  whose 
bases  are  5  and  8,  and  altitudes  12  and  6,  respectively  ? 

15.  Three  spheres  of  lead  whose  diameters  are  3,  4,  and  5 
inches,  respectively,  are  melted  and  formed  into  a  single 
sphere.  Find  its  diameter,  having  given  that  the  volume 
of  a  sphere  varies  as  the  cube  of  its  diameter. 

16.  The  volume  of  a  cone  of  revolution  varies  jointly  as 
its  altitude  and  the  square  of  the  radius  of  its  base.  If  the 
volume  of  a  cone  whose  altitude  is  3  and  radius  of  base  5 
is  784,  what  will  be  the  radius  of  the  base  of  a  cone  whose 
volume  is  47^  and  altitude  5  ? 

17.  If  5  men  in  6  weeks  earn  $57,  how  many  weeks  will 
it  take  4  men  to  earn  $  7G,  it  being  given  that  the  amount 
earned  varies  jointly  as  the  number  of  men,  and  the  number 
of  weeks  during  which  they  work  ? 

18.  If  the  volume  of  a  cylinder  of  revolution  varies  jointly 
as  its  altitude  and  the  square  of  its  radius,  what  will  be  the 
radius  of  a  cylinder,  whose  altitude  is  18,  equivalent  to  the 
sum  of  two  cylinders  whose  altitudes  are  5  and  12,  and  radii 
6  and  9,  respectively  ? 

19.  If  the  illumination  from  a  source  of  light  varies 
inversely  as  the  square  of  the  distance,  how  much  farther 
from  a  candle  must  a  book,  which  is  now  15  inches  off,  be 
removed,  so  as  to  receive  just  one-third  as  much  light  ? 


ARITHMETICAL   PROGRESSION.  285 


XXVIL    ARITHMETICAL  PROGRESSION. 

414.  An  Arithmetical  Progression  is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding  by  adding  a 
constant  quantity  called  the  common  difference. 

Thus,  1,  3,  5,  7,  9,  11,  ...  is  an  increasing  arithmetical 
progression,  in  which  the  common  difference  is  2. 

Again,  12,  9,  6,  3,  0,  —3,  ...  is  a  decreasing  arithmetical 
progression,  in  which  the  common  difference  is  —  3. 

415.  Given  the  first  term.,  a,  the  common  difference,  d,  and 
the  number  of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is 

a,  a  +  d,  a  -{-2d,  a-\-3d,  •  •  •  • 

It  will  be  observed  that  the  coefficient  of  d  in  any  term  is 
one  less  than  the  number  of  the  term.  Hence,  in  the  nth 
or  last  term,  the  coefficient  of  d  will  be  n  —  1. 

That  is,  I  =  a-\-(n  -  l)fL  (I.) 

416.  Given  the  first  term,  a,  the  last  term,  I,  and  the  number 
of  terms,  n,  to  find  the  sum  of  the  series,  S. 

^  =  a  +  (a  +  d)  +  (a  +  2d)  +  •••  +  (Z  -  d)  +  Z. 
Writing  the  series  in  reverse  order, 

S  =  I  -\-  (I  -  d)  -h  {I  -  2d)  -\-  ■■■  -{- {a -{- d)  +  a. 
Adding  these  equations,  term  by  term, 
•      2;S'  =  (a+0  +  (a+0  -f-  (a  +  0  +  •  •  •  +  (a+0  +  («  +  0 

=  n(a  +  0- 
Therefore,  ^  =  '^  (a  + 1) •  (H.) 

417.  Substituting  in  (II.)  the  value  of  I  from  (I.),  we  have 

>S  =  |[2a+(n-l)d]. 


286 


COLLEGE   ALGEBRA. 


EXAMPLES. 

418.  1.  In  the  series  8,  5,  2,  -1,  -4,  •••  to  27  terms, 
find  the  last  term  and  the  sum. 

In  this  case,  a  =  8,  cZ  =  —  3,  n  =  27. 
Substituting  in  (I.)  and  (II.), 

Z  =  8  +  (27-l)(-3)  =  8-78  =  -    70. 

^  =  ?I(8-70)=27  X  (-31)  =-837. 

2 

Note.  The  common  difference  may  be  found  by  subtracting  any 
term  of  the  series  from  the  next  following  term.  Thus,  m  the  pro- 
gression 

5         1^  1  J  1      5  11 

_.   —-,  —2,  •",  we  have  a  = -  = r" 

3         0  b      3  b 

In  each  of  the  following,  find  the  last  term  and  the  sum 
of  the  series : 

2.  3,  11,  19,  •••  to  19  terms. 

3.  _  5,  -  11,  -  17,  •••  to  22  terms. 

4.  _  69,  —  62,  —  55,  •••  to  IG  terms. 

5.  ^,  _§,_?,...  to  13  terms. 
4'       8'       2' 

6.  -,—,•••  to  25  terms. 

7.  _i  1   ...  to  38  terms. 

3' 2' 

8.  —-,—'-,•••  to  55  terms. 

4'       G' 

9.  — -,  -•^,  ...  to  17  terms 

5'       2' 


10.    2a -5&,  7a -2  6,  •••  to  9  terms. 


11.    ^  -  y^  ?/,...  to  10  terms. 


ARITHMETICAL   PROGRESSIOX.  287 

419.  If  any  tliree  of  tlie  five  elements  of  an  arithmetical 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  known  values  in  the  fundamental  formulae  (I.) 
and  (11.),  and  solving  the  resulting  equations, 

5  5 

1.  Given  a  =  —  *-,  7i  =  20,  S  =  —  -;  find  d  and  L 

3  3 

Substituting  the  given  values  in  (I.)  and  (II.),  we  have 

Z  =  -^  +  19rf.  (1) 

3 

3  \     3       J'  6  3  ^  ^ 

From  (2),  ;  =  ---  =  -. 
^  ^  3      6      2 

SulBstituting  in  (1), 

^  =  —  1^  +  19  cZ ;  whence,  d  =  — 
2  3  6 

2.  Given  d  =  -  3,  l  =  -39,  S  =  -264.;  find  a  and  n. 
Substituting  in  (I.)  and  (II.), 

-39  =  a  +  (7i-l)(-3);  or  a  =  37i-42.  (1) 

-  264  =  ^  (a  -  39)  ;  or  an  -  39  w  =  -  528.  (2) 

Substituting  the  value  of  a  from  (1)  in  (2), 
3^2  _42u- 3971  =  -528, 
or,  n-  —  27  n  =  —  176. 


wi..  ..  27±V729-704      27  ±  5      -.^        .. 

Whence,  n  = = =  16  or  11 

2  2 

Substituting  in  (1), 

a  =  48  -  42  or  33  -  42  =  6  or   -  9. 
'riierefore,  a  =  Cj  and  n  =  16  ;  or,  a  =  —  9  and  n  =  11. 


288  COLLEGE    ALGEBRA. 

Note  1.   The  significance  of  the  two  answers  is  as  follows : 

If  a  =  6  and  n  =  16,  the  series  is 

6,  3,  0,  -3,  -0,  -9,  -12,  -15,  -18,  -21,  -24,  -27,  -30, 
_33, -3G, -39. 

If  a  =  —  9  and  7i  =  11,  tlie  series  is 

_9,  _i2,  -15,  -18,  -21,  -24,  -27,  -30,  -33,  - 3G,  -39. 

-     In  each  of  these  tlie  last  term  is  —  39,  and  the  siun  —  2G4. 


3.    Given  a  =  -,  d  = ,  S  —  —  '-:  find  I  and  ?i. 

3  12  2 

Substituting  in  (I.)  and  (II.), 

i  =  |  +  (»-i)(-i);o.-'  =  ^-         (1) 

Substituting  the  value  of  I  from  (1)  in  (2), 


4 

Solving  this  equation,  n  =  12  or  —  3. 

The  second  value  is  inapplicable,  for  the  number  of  terms 
in  a  progression  must  be  a  positive  integer. 

Substituting  the  value  n  =  12  in  (1), 

,5-12  7 


12  12 


Therefore,  I  —  ——  and  n  =  12. 

Note  2.   A  negative  or  fractional  value  of  n  is  inapplicable,  and 
must  be  rejected  together  with  all  other  values  dependent  upon  it. 

Note  3.    The  series  con-espondiug  to  the  value  n  =  12  in  Ex.  3  is 

1     1     i     i_     0     -  ^      _1     _1     _1     _^     _1     _J. 
3'    4'    G'    12'      '        12'        G'        4'        3'        12'        2'        12* 


ARITHMETICAL   PROGRESSION.  289 

It  will  be  observed  that  if  we  count  backwards  three  terms,  beginnmg 
with  the  last  term,  we  have  a  series  —  :^,  —-,——,  whose  sum  is  —  ^- 

This  series  has  the  same  common  difference,  last  term,  and  sum  as 
the  given  series  ;  but  it  has  not  the  same  first  term,  and  hence  does  not 
satisfy  aU  the  given  conditions. 

It  will  always  be  found  that,  when  a,  d,  and  8  are  given,  and  one 
of  the  values  found  for  n  is  a  positive  and  the  other  a  negative  integer, 
the  series  obtained  by  counting  backwards  from  the  last  term  of  the 
series  corresponding  to  the  positive  value  of  n,  as  many  terms  as  are 
indicated  by  the  negative  value  of  n,  will  have  its  sum  equal  to  the 
given  sum. 

In  like  manner,  it  will  be  fomid  that,  when  Z,  d^  and  S  are  given, 
and  one  of  the  values  fomid  for  n  is  a  positive  and  the  other  a  negative 
integer,  the  series  obtained  by  counting  forwards  from  the  first  term  of 
the  series  corresponding  to  the  positive  value  of  n,  as  many  terms  as 
are  indicated  by  the  negative  value  of  ?«,  will  have  its  sum  equal  to  the 
given  sum. 

EXAMPLES. 


4. 

Given  d  =  5,  1  =  11,  n  =  15 ;  find  a  and  S. 

5. 

Given  d--4.,n  =  20,  >S'  =  -  620 ;  find  a  and 

6. 

Given  a  =  —  9,  7i  =  23,  1=  57 ;  find  d  and  S. 

7. 

Given  a  =  —  5,  n  =  29,  S  =  —  2175 ;  find  d  and 

8. 

Given  a  =  -,  Z  =  — ,  ^  =  ^ ;  find  d  and  n. 

9. 

Given   l  =  —  -,n=19,S  =  0',  find  a  and  d. 
o 

10. 

1           n^          o 
Given  d  =  ^,  S  =  -^,  a  =  ^',  find  Z  and  n. 
12           3           u 

11. 

Given  a  =  l,  l  =  - ^,  d  =  -^;  find  n  and  S. 

12. 

Given  d  =  ^,  n  =  17,  S  =  17 -,  find  a  and  I 

13. 

Given  I  =  ^^'',  d  =  ^^,  S=^^^  :  find  a  and  n. 
15 '         15'           15  ' 

LOO  COLLEGE   ALGEBRA. 

14. .  Given   ?  =  —  5J,  n  =  21,  *S'  =  —  38-} ;  find  a  and  d. 

15.  Given  a  =  —  5,  /  =  —  47,  S  =  —  1118 ;  find  d  and  n. 

0()3 

16.  Given  a  =  G,  n  =  14,  /S  =  —  ^^  ;  find  d  and  Z. 

o 

17.  Given  J  =  —  -,  d  = 3,  aS'  = ;  find  a  and  n. 

3  15  o 

18.  Given  a  =  -  ?  d  =  ^,  >&'  =  120  :  find  71  and  I. 

4'  4' 

From  (I.)  and  (II.)  general  formulse  for  the  solution  ol 
cases  like  the  above  may  be  readily  derived. 

^  19.    Given  a,  d,  and  S ;  derive  the  formula  for  n. 

Substituting  the  value  of  I  from  (I.)  in  (II.), 

2S=  n\2a  +  {n  -  l)f/],  or  dn'  +  (2a  -  d)n  =  2S. 

This  is  a  quadratic  in  n,  and  may  be  solved  by  the  method 
of  Art.  345. 

Multiplying  by  4  d,  and  adding  {2  a  — dy  to  both  members, 

4cZV+  4d(2a  -d)n  +  (2a-dy  =  8dS  +  (2 a  -  df. 

Extracting  the  square  root, 

2dn  +  2a-d  =  ±  VSdS  +  {2a-dy. 


,^,  d  —  2a  ± VSdS  +  (2a  —  dy 

Whence,  n  = -^-^^ ^• 

2d 

20.  Given  a,   I,  and  n ;  derive  the  formula  for  d. 

21.  Given  a,  n,  and  S ;  derive  the  formulse  for  d  and  /. 

22.  Given  d,  n,  and  S;  derive  the  formulas  for  a  and  I. 

23.  Given  a,  d,  and   l;  derive  the  formulae  for  n  and  S. 

24.  Given  d,   I,  and  n ;  derive  the  formulse  for  a  and  S. 

25.  Given  I,  n,  and  S ;  derive  the^ormuhie  for  a  and  d. 

26.  Given  «,  d,  and  S ;  derive  the  formula  for  /. 


ARITHMETICAL   PROGRESSION.  291 

27.  Given  a,   I,  and  S ;  derive  the  formulas  for  d  and  n. 

28.  Given  d,   I,  and  S ;  derive  tlie  formula3  for  a  and  n. 

jj  420.  To  insert  any  number  of  arithmetical  means  between 
two  given  terms. 

Let  it  be  required,  for  example,  to  insert  5  arithmetical 
means  between  3  and  —  5. 

This  means  that  we  are  to  find  an  arithmetical  progression 
of  7  terms,  whose  first  term  is  3,  and  last  term  —  5. 

Putting  a  =  3,  ?  =  —  5,  and  n  =  7,  in  (I.),  we  have 

—  5  =  3  +  G  d,  or  d  =  —  -. 
3 

Hence  the  required  series  is 

Q     5     1         ^  7         11  w 

^'    3'    3'    -^'    -3'    --3'    -'■ 

421.  Let  X  denote  the  arithmetical  mean  between  a  and  b. 
Then,  by  the  nature  of  the  progression, 

X  —  a  =  b  —  X,  or  2x=  a  -{-b. 

Whence,  x  =  "  +  ^. 

2 

That  is,  tJie  arithmetical  mean  between  two  quantities  is 
equal  to  one-half  their  sum. 

EXAMPLES. 

422.  1.  Insert  6  arithmetical  metins  between  3  and  8. 

2.  Insert  8  arithmetical  means  between  -  and • 

2  10 

3.  Insert  7  arithmetical  means  between and  — 

2  2 

4.  Insert  8  arithmetical  means  between  —  -  and  —  5. 

4 


292  COLLEGE   ALGEBRA. 

3 

5.  Insert  9  arithmetical  means  between  -  and  —11. 

6.  If  m  arithmetical  means  are  inserted  between  a  and  b, 
what  are  the  first  and  last  means  ? 

7.  Find  the   number  of  arithmetical  means  betAveen   ^ 

9  9  '' 

and ,  when  the  sum  of  the  first  two  is  — 

7  35 

Find  the  arithmetical  mean  between  : 

8.  21  and  -If.  9.    (a  +  &)'and   -{a -by. 

10.    ^^  +  ^  and  '±^- 
a  —  b  a  -\-b 


PROBLEMS. 

423.    1.  The  sixth  term  of  an  arithmetical  progression  is 

-,  and  the  fifteenth  term  is  — .     Find  the  first  term. 
6  3 

By  Art.  415,  the  sixth  term  is  a  +  5  d,  and  the  fifteenth  term  is 
a+lid;  hence, 

(«+    5d  =  |  (1) 

■^  I  a+14cZ=  Y  (2) 

Subtracting  (1)  from  (2), 

Substituting  iu(l),  a +■-=-_;  whence,  a  =  —  ^- 

2      0  o 

2.  Find  four  quantities  in  arithmetical  progression  such 
that  the  product  of  the  extremes  shall  be  45,  and  the  product 
of  the  means  77. 

Let  the  quantities  he  x  —  Zy,  x  —  y,  x+  ?/,  and  x  +  Sy. 


Then  by  the  conditions,  < 


x2_02/2=45. 


2-      »/2=7/ 


Solving  these  equations,  jc  =  0,  y—±2;  or,  x  =  — 0, 7y=±2(Art.  307). 
Hence  the  quantities  are  3,  7, 11,  and  15  ;  or,  —  3,  —  7,  —  11,  and  — 15. 


ARITHMETICAL  PROGRESSION.  293 

Note.  In  problems  like  the  above  it  is  convenient  to  represent  the 
imknowu  quantities  by  symmetrical  expressions.  Thus  if  five  quanti- 
ties had  been  required,  we  should  have  represented  them  by  aj  —  2  ?/, 
x  —  y,  X,  X  +  y,  and  x  +  2  ?/. 

—  3.    Find  the  sum  of  the  even  integers  beginning  with  2 
and  ending  with  500. 

-T    4.    The  7th  term  of  an  arithmetical  progression  is  27, 
and  the  13th  term  is  —  3.     Find  the  21st  term. 

5.  Find  four  numbers  in  arithmetical  progression  such 
that  the  sum  of  the  first  two  shall  be  12,  and  the  sum  of  the 
last  two  —  20. 

6.  The  19th  term  of  an  arithmetical  progression  is 
9  a  —  2  &,  and  the  31st  term  is  13  a  —  8  6.  Find  the  sum  of 
the  first  thirteen  terms. 

7.  Find  the  sum  of  the  first  n  positive  integers  which 
are  multiples  of  7. 

8.  Find  four  integers  in  arithmetical  progression  such 
that  their  sum  shall  be  24,  and  their  product  945. 

9.  Find  the  sum  of  all  positive  integers  of  three  digits 
Avhich  are  multiples  of  11. 

10.  The  7th  term  of  an  arithmetical  progression  is  —  i, 
the  16th  term  is  2i,  and  the  last  term  is  6|-.  Find  the 
number  of  terms. 

11.  Find  five  quantities  in  arithmetical  progression  such 
that  the  sum  of  the  first,  third,  and  fourth  shall  be  3,  aiid 
the  product  of  the  second  and  fifth  —  8. 

12.  A  body  falls  16 Jj  feet  the  first  second,  and  in  each 
succeeding  second  32|-  feet  more  than  in  the  next  preceding 
one.     How  far  will  it  fall  in  16  seconds  ? 

13.  Find  three  quantities  in  arithmetical  progression  such 
that  the  sum  of  the  squares  of  the  first  and  third  exceeds 
the  second  by  123,  and  the  second  exceeds  one-third  of  the 
first  by  6. 


294  COLLEGE    ALGEBRA. 

14.  A  man  travels  3  miles  the  first  day,  6  miles  the  second 
day,  9  miles  the  third  day,  and  so  on.  After  he  has  travelled 
a  certain  number  of  days,  he  finds  that  his  average  daily 
distance  is  46|-  miles.  How  many  days  has  he  been  travel- 
ling? 

15.  The  mth  term  of  an  arithmetical  progression  is  p, 
and  the  nth  term  is  q.     What  is  the  (m  +  ?i)th  term  ? 

16.  Find  the  number  of  arithmetical  means  between  1 
and  31,  when  the  seventh  mean  is  to  the  one  before  the  last 
as  5  is  to  9. 

17.  After  A  had  travelled  for  4^  hours  at  the  rate  of  5 
miles  an  hour,  B  set  out  to  overtake  -  him,  and  travelled  3 
miles  the  first  hour,  31  miles  the  second  hour,  4  miles  the 
third  hour,  and  so  on.     In  how  many  hours  will  B  overtake 

A  ?      '^'    '' ''  '■'.''  ■   .    ^ 

18.  Find  three  numbers  in  arithmetical  progression  such 
that  the  sum  of  their  squares  is  347,  and  one-half  the  third 
number  exceeds  the  sum  of  the  first  and  second  by  4|^. 

19.  If  a  person  saves  f  100  a  year,  and  puts  this  sum  at 
simple  interest  at  5  per  cent  at  the  end  of  each  year,  to  how 
much  will  his  property  amount  at  the  end  of  20  years  ? 

20.  The  digits  of  a  number  of  three  figures  are  in  arith- 
metical progression ;  the  first  digit  exceeds  the  sum  of  the 
second  and  third  by  1 ;  and  if  594  is  subtracted  from  the 
number,  the  digits  will  be  inverted.     Find  the  number. 

21.  There  are  two  sets  of  numbers,  each  consisting  of 
three  terms  in  arithmetical  progression  whose  sum  is  15. 
The  common  difference  of  the  first  set  exceeds  by  vinity  the 
common  difference  of  the  second  set ;  and  the  product  of 
the  first  set  is  to  the  product  of  the  second  as  7  is  to  8. 
Required  the  numbers. 


GEOMETRICAL  PROGRESSION.  295 


XXVIII.     GEOMETRICAL   PROGRESSION. 

424.  A  Geometrical  Progression  is  a  series  of  terms,  each 
of  which,  is  derived  from  the  preceding  by  multiplying  by  a 
constant  quantity  called  the  ratio. 

Thus,  2,  6,  18,  54,  1G2,  •••  is  an  increasing  geometrical 
progression  in  which  the  ratio  is  3. 

Again,  9,  3,  1,  -,  -,  •••  is  a  decreasing  geometrical  pro- 
3   9  ^ 

gression  in  which  the  ratio  is  -  • 

Negative  values  of  the  ratio  are  also  admissible ;  thus, 
—  3,  6,  —  12,  24,  —  48,  ...  is  a  geometrical  progression  in 
which  the  ratio  is  —  2. 

425.  Given  the  first  term,  a,  the  ratio,  r,  and  the  number 
of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is  a,  ar,  ar^,  ar^,  .... 

It  will  be  observed  that  the  exponent  of  r  in  any  term  is 
one  less  than  the  number  of  the  term.  Hence,  in  the  7ith  or 
last  term,  the  exponent  of  r  will  be  w  —  1. 

That  is,  l  =  ar"^\  (I.) 

426.  Given  the  first  term,  a,  the  last  term,  I,  and  the  ratio, 
r,  to  find  the  sum  of  the  series,  S. 

S  =   a-{-  ar  -\-ar^  +  '••  -\-  a?-"^^  +  ar"'~-\-  ar'*^^ 
Multiplying  each  term  by  r, 

rS  =  ar  +  aj"  +  a?-^  +  •  •  •  +  or""-  +  a?-""^  +  ar". 
Subtracting  the  first  equation  from  the  second, 

rS  —  S  =  ar"  —  a;  or,  S  =  — ~    • 
r  —  1 

But  by  (I.),  Art.  425,  rl  =  ar'\ 

Therefore,  6^  =  !izL^.  (H.) 


296  COLLEGE   ALGEBRA. 


EXAMPLES. 

427.    1.  In  the  series  3,  1,  -,  •••  to  7  terms,  find  the  last 
term  and  the  sum. 

In  this  case,  a  =  3,  r  =  -  n  =  7.     Substituting  in  (I.)  and 
(11.),  ^ 


2186 
729       1093 

-  3-     -i 

2          243 
3 

Note.    The  ratio  may  be  found  by  dividing  any  term  by  tlie  next 
preceding  term. 

2.    In  the  series  —  2,  6,  —  18,  54,  ...  to  8  terms,  find  the 
lasf  term  and  the  sum. 

In  this  case,  a  =  —  2,  r=  — -  =  —  3,  n  =  8.     Hence, 

^  =  -2(-3)"  =  -2  x(- 2187)  =  4374. 

^  -3x4374 -(-2)  ^  -  13122  +  2  ^  ^ogQ 
-3-1  -4 

In  each  of  the  following,  find  the  last  term  and  the  sum 
of  the  series : 


3. 

1,  3,  9,  •••  to  9  terms. 

4. 

6,  4,  ^,  •••  to  7  terms. 

5. 

-2,  10,  -50,  ...  to  5  terms. 

6. 

-3,  1  -|  ...to  8  terms. 

7. 

-?,  -5,  -10,  ...  to  10  terms. 

GEOMETRICAL   PROGRESSION.  297 


25 
8.    ^^,  —  5,  2,  •••  to  6  terms. 

2'        '    ' 


1   1   _3 

3'  2'      4' 


to  7  terms. 


10.    5,  A,  2^    ...  to  5  terms. 


11.    - 


4  16  64 
3  3 


-,  —  6,  •  •  •  to  6  terms. 

2' 


■  428.  If  any  three  of  the  five  elements  of  a  geometrical 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  known  values  in  the  fundamental  formulae  (I.) 
and  (II.),  and  solving  the  resulting  equations. 

But  in  certain  cases  the  operation  involves  the  solution 
of  an  equation  of  a  degree  higher  than  the  second ;  and  m 
others  the  unknown  quantity  appears  as  an  exponent/ the 
solution  of  which  form  of  equation  can  usually  only  be 
effected  by  aid  of  logarithms  (Art.  519). 

In  all  such  cases  in  the  present  chapter,  the  equations 
may  be  solved  by  inspection. 

1 .    Given  a  =  —  2,  n  =  5,  I  =  —  32;  find  r  and  S. 
Substituting  the  given  values  in  (I.),  we  have 

-  32  =  -  2r*;  whence,  r*  =  16,  or  r  =  ±  2. 
Substituting  in  (II.), 
If  r  =      2,  S  =  2(^-32) -(-2)  ^  _  54  _,_  2  =  _  62. 

If  r  =  -2,  ^^(-2)(^32_)-l-2^^64+_2^_,, 

-2-1  -3 

Therefore,  r  =  2  and  ^=  -62  ;  or,  r  =  -  2  and  S=  -22, 
Note  1.  The  significance  of  the  two  answers  is  as  follows : 

If  r=z     2,  the  series  is  —2,  —4,  —8,  —16,  —32,  in  which  the  sum  is  —02. 

K  r  —  —  2,  the  series  is  —2,     4,  —  8,     16,  —32,  in  whicli  the  sum  is  —22. 


298  COLLEGE  ALGEBRA. 

2.    Given  a  =  3,  r  =  -|^  =  ^ ;  find  n  and  I 

Substituting  in  (II.), 

-1^-3 
1640  ^      3  ^  Z  +  9 

729        _l_i  4    ■ 

3 

Whence,      ^  +  9  =  ^;  or,  Z  = -^. 

Substituting  in  (I.), 

Whence,  by  inspection, 

w  —  1  =  7,  or  ?*  =  8. 

EXAMPLES. 
-  3.    Given  r  =  2,  w  =  9,  Z  =  256 ;  find  a  and  S. 
4.    Given  a  =  -  2,  ?i  =  6,  Z  =  2048 ;  find  r  and  S. 

9  '^O'lO 

^     5.    Given  r  =  ^,  w  =  7,  >S  =  ^^ ;  find  a  and  Z. 

3  24o 

6.    Given  a  =  2,  r  =  -  -,  Z  =  -  — - ;  find  n  and  aS'. 
2  256 

1  /*047 

—   7.    Given  r=^,n=  11,  aS  =  7^7-7^ ;  find  a  and  Z. 

2  2U4o 

8.    Given  a  =  |  ?i  =  9,  Z  =  ^-^ ;  find  r  and  ^. 

1  511 

^      9.    Given  a  =  -  8,  Z  =  -  -^,  ^  =  -  ':^;  find  r  and  n. 

10.  Given  a  =  -,  r  =  -  ^,  /S  =  -^  ;  find  Z  and  w. 

4  3  162 

11.  Given  I  =  192,  r  =  -2,  S  =  129;  find  a  and  n. 

12.  Givena  =  -|z  =  -^,  ^  =  -|||;  and  r  and  n. 


GEOMETRICAL  PROGRESSION.  299 

From  (I.)  and  (II.)  general  formulae  may  be  derived  for 
the  solution  of  cases  like  tlie  above. 

13.  Given  a,  r,  and  S ;  derive  the  formula  for  I. 

14.  Given  a,   Z,  and  S ;  derive  the  formula  for  r. 

15.  Given  r,   I,  and  S ;  derive  the  formula  for  a. 

16.  Given  r,  n,  and   I ;  derive  the  formulae  for  a  and  S. 
-    17.  Given  r,  n,  and  yiS*;  derive  the  formulae  for  a  and  I. 

18.    Given  a,  n,  and   I ;  derive  the  formulae  for  r  and  S. 

Note  2.  If  the  given  elements  are  n,  I,  and  5,  equations  for  a  and 
r  may  be  found,  but  there  are  no  definite  fonnulce  for  their  values. 
The  same  is  the  case  when  the  given  elements  are  a,  n,  and  S. 

The  general  formulae  for  n  involve  logarithms  ;  these  cases  are  dis- 
in  Art.  519. 


"^  429.  The  limit  (Art.  209)  to  which  the  sum  of  the  terms 
of  a  decreasing  geometrical  progression  approaches,  as  the 
number  of  "terms  increases  indefinitely,  is  called  the  sum  of 
the  series  to  infinity. 

The  value  of  S  in  formula  (II.),  Art.  426,  may  be  written 
ci      a  —  rl 

It  is  evident  that,  by  sufficiently  continuing  a  decreasing 
progression,  the  absolute  value  of  the  last  term  may  be  made 
less  than  any  assigned  positive  quantity,  however  small. 

Hence,  as  the  number  of  terms  increases  indefinitely,  I 
approaches  the  limit  0,  and  therefore  rl  approaches  the 
limit  0. 

Then  the  fraction  ''''  ~^    approaches  the  limit 

1  —r  1  —  r 

That  is,  the  sum  of  a  decreasing  geometrical  progression 
to  infinity  is  given  by  the  formula 

S  =  ^^-  (III.) 

1  — r 


300  COLLEGE   ALGEBRA. 


EXAMPLES. 


S   16 

1.    Find  the  sum  of  the  series  4, ,  — ,  •••  to  infinity. 

o    9 

2 

In  this  case,  a  =  4,  r  = 

3 


Substituting  in  (III.),  S 


4     _12 

1  +  ^       5- 


Note.    This  signifies  that,  the  greater  the  number  of  terms  taken, 

*  1'' 

the  more  nearly  does  their  sum  approach  to  -^  ;  but  no  matter  how 

5 

many  terms  are  taken,  tlie  sum  will  never  exactly  equal  this  value. 
Find  the  sum  of  the  following  to  infinity : 


2. 

3,.|.... 

6. 

7    21     63 
4'  32'  256'  " 

•• 

3. 

16,  -4,  1,  .... 

7. 

2       1     5  * 
5'      3'  18'  " 

4. 

_1    1    _1  .... 

'  5'       25' 

8. 

1        1 

8'       18' 

2 
'81' 

5. 

5       10        20 

3'        9'       27'"'" 

9. 

40     .5    5 

49'       7'  8'  ■■ 

430.    To  find  the  value  of  a  repeating  decimal. 

This  is  a  case  of  finding  the  sum  of  a  geometrical  pro- 
gression to  infinity,  and  may  be  solved  by  the  formula  of 
Art.  429. 

1.    Find  the  value  of  .85151  .... 

.85151  ...  =  .8  +  .051  +  .00051  H 

The  terms  after  the  first  constitute  a  decreasing  geometri- 
cal progression,  in  which  a  =  .051  and  r  —  .01. 


GEOMETRICAL   PROGRESSION.  30I 


Sabstitutin 

ig  in  (III.), 

^         .051 
^  =  1-.01 

.051 
.99 

_  51  _ 

990 

17 

'330' 

Hence  the 

value  of  the 

given  decimal 

is 

8 
10 

^1'   = 
330 

281 
330 

EXAMPLES. 
Find  the  values  of  the  following : 

2.  .7272....  4.    .G9444....  6.    .11003003.... 

3.  .407407  ....  5.    .58686  ....  7.    .922828  .... 

431.  To  insert  any  number  of  geometrical  means  between 
tivo  given  terms. 

Let  it  be  required,  for  example,  to  insert  5  geometrical 

1*^8 

means  between  2  and  -^^• 

729 

This  means  that  we  are  to  find  a  geometrical  progression 

of  7  terms,  whose  first  term  is  2,  and  last  term  -^^^ 

729 

128 
Putting  a  =  2,  1  =  - — ,  and  n=  7,  in  (I.),  we  have 

— ^  =  2?"^ ;  whence,  r^  = ,  and  r  =  ±  -• 

729  729  3 

Hence,  the  required  result  is 

9    ±^    §.    ±1^    32        J>4    128 
3'  9'       27'  81'       243'  729' 

432.  Let  X  denote  the  geometrical  mean  between  a  and  b. 

Then  by  the  nature  of  the  progression, 

X      b  ■>        , 

-  =  -,  or  or  =  ab. 
a      X 

Whence,  x  =  Va6. 


302  COLLEGE   ALGEBRA. 

That  is,  the  geometriccd  mean  between  two  quantities  is  equal 
to  the  square  root  of  their  product. 

EXAMPLES. 

433.    1.  Insert  6  geometrical  means  between  -  and  — 

6  3 

2.  Insert  4  geometrical  means  between  4  and  —  972. 

3.  Insert  5  geometrical  means  between  —  2  and  — 128. 

2  125 

4.  Insert  3  geometrical  means  between and —• 

7 

5.  Insert  4  geometrical  means  between and  3584. 

243  2 

6.  Insert  7  geometrical  means  between  - —  and  -^« 

7.  If  m  geometrical  means  are  inserted  between  a  and  h 
what  is  the  first  mean  ? 

Find  the  geometrical  mean  between  : 

8.  llf  and  2f 

9.  4  a.-^  -f  12  xy  +  9  y-  and  4  x-  —  12  xy  +  9  y\ 

10.    «'-^^^  and  '''+''^- 
ah  +  ly"  ah  -  W 


PROBLEMS. 

434.  1.  Find  three  numbers  in  geometrical  progression, 
such  that  their  sum  shall  be  14,  and  the  sum  of  their  squares 
84. 

Let  the  numbers  be  a,  ar,  and  ar- ;  then,  by  the  conditions, 

^  a-\-  ar-\-ar'^      =  14.  (1) 

\  a?-  +  a2>-2  +  a2)-*  =  84.  (2) 

Dividing  (2)  by  (1),  a-  ar  +  af^  =  G.  (3) 

Subtractuig  (3)  from  (1),  2ar=8,  or  r  =  --  (4) 

a 


GEOMETRICAL   PROGRESSION.  303 


Substituting  iu(l), 

a 

Or, 

a^- 10a  =  -16. 

Solving  tliis  equation, 

a  =  8  or  2. 

Substituting  in  (4), 

-t-i- 

Therefore,  the  numbers 

are 

2,  4,  and  8. 

2.  The  fifth  term  of  a  geometrical  progression  is  48,  and 
the  eighth  term  is  —  384.     Find  the  first  term. 

3.  The  sum  of  the  first  and  second  of  four  numbers  in 
geometrical  progression  is  15,  and  the  sum  of  the  third  and 
fourth  is  60.     What  are  the  numbers  ? 

4.  Find  three  numbers  in  geometrical  progression,  such 
that  the  sum  of  the  first  and  second  is  20,  and  the  third 
exceeds  the  second  by  30. 

5.  The  fourth  term  of  a  geometrical  progression  is  —108, 
and  the  tenth  term  is  —  78732.     What  is  the  first  term  ? 

6.  The  elastic  power  of  a  ball,  which  falls  from  a  height 
of  100  feet,  causes  it  to  rise  .9375  of  the  height  from  which 
it  fell,  and  to  continue  in  this  way  diminishing  the  height 
to  which  it  will  rise,  in  geometrical  progression,  until  it 
comes  to  rest.     How  far  will  it  have  moved  ? 

7.  The  sum  of  four  numbers  in  geometrical  progression 
is  30,  and  the  quotient  of  the  fourth  number  divided  by  the 
sum  of  the  second  and  third  is  1^.     Find  the  numbers. 

8.  The  third  term  of  a  geometrical  progression  is  J^, 
and  the  sixth  term  is  -gf^.     Find  the  eighth  term. 

9.  Divide  the  number  39  into  three  parts  in  geometrical 
progression,  such  that  the  third  part  shall  exceed  the  first 
by  24. 

10.  The  product  of  three  numbers  in  geometrical  progres- 
sion is  G4,  and  the  sum  of  the  squares  of  the  first  and  third 
is  G8.     What  are  the  numbers  ? 


304  COLLEGE   ALGEBRA. 

11.  The  sum  of  three  numbers  in  arithmetical  progression 
is  12.  If  the  first  number  is  increased  by  5,  the  second  by 
2,  and  the  third  by  7,  the  results  form  a  geometrical  pro- 
gression.    What  are  the  numbers  ? 

12.  The  product  of  three  numbers  in  geometrical  progres- 
sion is  8,  and  the  sum  of  their  cubes  is  73.  What  are  the 
numbers  ? 

13.  Divide  f  700  between  A,  B,  C,  and  D,  so  that  their 
shares  may  be  in  geometrical  progression,  and  the  sum  of 
A's  and  B's  shares  equal  to  $  252. 

14.  There  are  four  numbers,  the  first  three  of  which  form 
an  arithmetical  progression,  and  the  last  three  a  geometrical 
progression.  The  sum  of  the  first  and  third  is  2,  and  of 
the  second  and  fourth  37.     What  are  the  numbers  ? 

15.  What  is  the  ratio  of  the  geometrical  progression,  the 
sum  of  Avhose  first  ten  terms  is  244  times  the  sum  of  its  first 
five  terms  ? 

16.  The  sum  of  the  first  three  terms  of  a  geometrical 
progression  is  one-fourth  the  sum  of  the  third,  fourth,  and 
fifth  terms ;  and  the  seventh  term  is  384.  Find  the  first 
term  and  the  ratio. 

17.  There  are  three  numbers  in  geometrical  progression 
whose  sura  is  57.  If  the  first  is  multiplied  by  i,  the  secoud 
by  I,  and  the  third  by  |-,  the  results  form  an  arithmetical 
progression.     What  are  the  numbers  ? 

18.  If  the  mth  term  of  a  geometrical  progression  is  ^>, 
and  the  nth.  term  is  q,  what  is  the  (m  +  w)th  term  ? 

19.  If  a,  h,  c,  and  d  are  in  geometrical  progression,  prove 
that 

(a  -cy  +  {h-  cY  +  ih-  df  =  (a  -  cZ)l 


HARMONICAL   PROGRESSION.  305 


XXIX.    HARMONICAL  PROGRESSION. 

435.  Quantities  are  said  to  be  in  Harmonical  Progression 

when  tlieir  reciprocals  form  an  arithmetical  progression. 

Thus,   1,   -,    -,  -,    -,   •••  are  in  harmonical   progression, 
3    5    7    9 
l)ecause  their  reciprocals,  1,  3,  5,  7,  9,  ...,  form  an  arith- 
metical progression. 

436.  Any  problem  in  harmonical  progression,  which  is 
susceptible  of  solution,  may  be  solved  by  taking  the  recipro- 
cals of  the  terms,  and  applying  the  formulae  of  the  arith- 
metical progression. 

There  is,  however,  no  general  method  for  finding  the  sum 
of  the  terms  of  a  harmonical  series. 

437.  If  any  three  consecutive  terms  of  a  harmonical  series 
are  taken,  the  first  is  to  the  third  as  the  first  minus  the  second 
is  to  the  second  minus  the  third. 

Let  the  terms  be  a,  b,  and  c. 

Then  since  -,  -,  and  -  are  in  arithmetical  progression, 
ah  c 

1  _  1  ^  1  _  1^ 

c       b      h      a 

b  —  c  _  a  —  b 

be  ab 

Mviltiplying  both  members  by  -- — ,  we  have 
b  —  c 

a  _a  —  b 

c      b  —  c 

438.  Let  X  denote  the  harmonical  mean  between  a  and  b. 

Then  -  is  the  arithmetical  mean  between  -  and     • 
X  a  b 


306  COLLEGE   ALGEBRA. 


Whence,  by  Art.  421, 
Therefore, 


1      a      b      a+b 


X  2  2ab 

2  ah 

X  = 

a  +  h 


439.   Let  A,  G,  and  II  denote  the  arithmetical,  geonietri 
cal,  and  harmonical  means,  respectively,  between  a  and  h. 


Then  by  Arts.  421,  432,  and  438, 

and  //  = 

a  +  h 


^  =  ^^+-^(^  =  Vr^,and//=  2«& 


But,  — ■ —  X =  ah  —  i^aby. 

2  a+b  ^         ^ 


Whence,  A  K  11=  G-,  or  G  =  VA  x  H. 

.  That  is,  tJie  geometrical  mean  between  two  quantities  is  also 
the  geometrical  mean  hetioeen  their  arithmetical  and  harmonical 
means. 

440.   Let  a  and  b  be  two  positive  real  numbers. 

By  Art.  439,  the  positive  value  of  their  geometrical  mean 
is  intermediate  in  value  between  their  arithmetical  and 
harmonical  means. 

B  t  g  +  &       2 ah  _  (a  +  b)-  —  4ft6 

2         a  +  b"       2{a  +  b) 

^a'-2ab  +  b-^  (a-b)\ 
2{a  +  b)  2(a  +  &)' 

a  Xiositive  quantity. 

Hence,  of  the  three  means,  the  arithmetical  is  the  greatest, 
the  geometrical  next,  and  the  harmonical  the  least. 


to  3G  terms,  find  the 


EXAMPLES: 

441.   1. 

L. 

9     9 

the  series  2,  ",  ",  • 

iast  ten  11. 

HARMONICAL   PROGRESSION.  307 

Taking  the  reciprocals  of  the  terms,  we  have  the  arith- 

,.    ,        .13    5 

metical  series  -,  -,  -,  •••  • 

Zi    A    Li 

In  this  case  a  =  -,  cZ  =  1,  and  n  =  36. 
Substituting  in  (I.),  Art.  415,  we  have 

Z  =  |  +  (36-l)xl=^. 

2 

Then  —  is  the  last  term  of  the  given  harmonical  series. 

2.  Insert  5  harmonical  means  between  2  and  —  3. 

We  have  to  insert  5  arithmetical  means  between  -  and • 

2  3 

Substituting  a  =  -,  1  = ,  and  n  =  7,  in  (I.),  Art.  415, 

11  ^ 

--  =  -  +  6fZ:  or,  d=-— . 
3      2''  36 

Then  the  arithmetical  series  is 

1     13     2     J_     _J^     _1_     _1 

2'    36'    9'    12'        18'        36'        3* 

Therefore  the  required  harmonical  series  is 

2,    ?^    9     ^2,    _18,    _§6     _3. 
'    13'    2'        '  '         7' 

Find  the  last  terms  of  the  following : 

3.  ?,  A  1,  ...  to  11  terms.        4.    ?  -,  1,  -  to  17  terms= 
4  11   6  5  7 

5         5 

5. ,  10,  -,  •••  to  26  terms. 

3         4 

6.  6,  -  2,  --,•••  to  23  terms. 

7.  --,  -— ,  -— ,  •••  to  31  terms. 

7        23        16 


308  COLLEGE   ALGEBRA. 


3  15 

8.  Insert  7  harinonical  means  between  -  and  — 

2  26 

9.  Insert  5  harmonical  means  between and . 

5  13 

8 

10.  Insert  6  harmonical  means  between  4  and  — —• 

19 

Find  the  harmonical  mean  between  : 

11.  If  and  -  IvV  12.    ^^+^  and  ^'^^. 

13.  Find  the  last  term  of  the  harmonical  series  a,  ^,  ... 
tan  terms. 

14.  If  m  harmonical  means  are  inserted  between  a  and  b, 
what  is  the  second  mean  ? 

15.  The  first  term  of  a  harmonical  series  is  x,  and  the 
second  term  is  y ;  continne  the  series  to  three  more  terms. 

16.  The  arithmetical  mean  between  two  numbers  is  —  \, 
and  the  harmonical  mean  is  42.     What  are  the  numbers  ? 

17.  The  fourth  term  of  a  harmonical  series  is  —  f,  and 
the  ninth  term  is  —  i.      What  is  the  seventh  term  ? 

18.  The  geometrical  mean  between  two  numbers  is  12, 
and  the  harmonical  mean  is  9f . "   What  are  the  numbers  ? 

19.  There  are  three  numbers  in  harmonical  progression 
whose  sum  is  ff.  If  the  second  and  third  numbers  are 
multiplied  by  5  and  16,  respectively,  the  three  numbers 
form  a  geometrical  progression.     What  are  the  numbers  ? 

20.  If  the  mth  term  of  a  harmonical  progression  is  2h  '^i^d 
the  nth.  term  is  q,  what  is  the  (m  +  «)th  term  ? 

21.  Prove  that  if  a  is  the  arithmetical  mean  between  b 
and  c,  and  b  the  geometrical  mean  between  a  and  c,  then  c 
is  the  harmonical  mean  between  a  and  b. 


THE   BINOMIAL   THEOREM.  309 

XXX.     THE  BINOMIAL  THEOREM. 

POSITIVE   INTEGRAL   EXPONENT. 

442.  The  Binomial  Theorem  is  a  formula  by  means  of 
which  any  power  of  a  binomial  may  be  expanded  into  a 
series. 

We  shall  consider  in  the  present  chapter  those  cases  only 
in  which  the  exponent  is  a  positive  integer. 

PROOF   OF   THE   THEOREM    FOR   A   POSITIVE   INTEGRAL 
EXPONENT. 

443.  By  actual  multiplication,  we  obtain : 
(a  +  x)^  =  a-  +  2  ax  +  x- ; 

(a  +  aj)  3  =  a"  +  3  a-.^•  +  3  ax^  +  x^ ; 

(a  +  x)  ^  =  a^  +  4  a^x  +  6  a-x^  +  4  ax^  +  x^ ;  etc. 

In  the  above  results  we  observe  the  following  laws : 

I.  The  number  of  terms  is  greater  by  1  than  the  exr 
ponent  of  the  binomial. 

II.  The  exponent  of  a  in  the  first  term  is  the  same  as 
the  exponent  of  the  binomial,  and  decreases  by  1  in  each 
succeeding  term. 

III.  The  exponent  of  x  in  the  second  term  is  1,  and  in- 
creases by  1  in  each  succeeding  term. 

IV.  The  coefficient  of  the  first  term  is  1 ;  and  of  the 
second  term,  is  the  exponent  of  the  binomial, 

V.  If  the  coefficient  of  any  term  is  multiplied  by  the 
exponent  of  a  in  that  term,  and  the  result  divided  by  the 
exponent  of  x  increased  by  1,  the  quotient  will  be  the 
coefficient  of  the  next  following  term. 


310  COLLEGE   ALGEBRA. 

444.  We  will  now  prove  by  induction  (Art.  114,  Note) 
that  these  laws  hold  for  any  positive  integral  power  of  a+x. 

Assume  the  laws  to  hold  for  (a  +  x)";  where  n  is  any 
positive  integer. 

Then,   (a  +  re)"  =  a"  +  na"-^x  +  ^^^^^— ^avV  H (1) 

Z 

Let  P,  Q,  and  R  denote  the  coefficients  of  the  terms  in- 
volving a"~''a;'',  a"~''"^ic''+\  and  a"~''~^a;'"+^,  respectively,  in  the 
second  member  of  (1)  ;  thus, 

(a  +  a;)"  =  a"  +  na'^-'^x  +  ••• 

+  Pa"-''a;'-+Qa"--^x'-+i-f  7?a"-'-V+2-| (2) 

Multiplying  both  members  by  a  +  x,  we  have 
ya  +  xY^^ 

=  a"+i  +  ncCx  +  •••  -f  Qa"-''a;'-+i  +  Pa"-''-ia;'-+-  +  ••• 
+    a"a;  +  .-.  +  Pa"-''a;'-+^  +  Qa""""-' x'-+2  +  ... 
=  a"+^  +  (n  +  1)  cC'x  +  •  •  • 

+  (P  +  Q)a«-'-x-'-+i  +  (Q  +  P)a"--i.T'-+2+...  .  (3) 

This  result  is  in  accordance  with  the  second,  third,  and 
fourth  laws  of  Art.  443. 

Since  the  fifth  law  of  Art.  443  is  assumed  to  hold  with 
respect  to  the  second  member  of  (2),  we  shall  have 

Q  =  -P<"-'-),a..J7i=<?<''-'--l>. 
?•  +  1  r  +  2 

Therefore, 

^    ^    Q{n-r-\)       Q{n  +  1) 

Q±Ji^ r  +  2         _      r  +  2     ^n-r 

P+Q         Q{r  +  1)    J   ^  Q(n  +  1)      r  +  2' 

?i  —  7"  ■  n  —  r 

Whence,  Q  +  /j  =  ( />  +  Q )  !izil". 

r-h2 


THE   BINOMIAL   THEOREM.  311 

But  n  —  r  is  the  exponent  of  a  in  that  term  of  (3)  whose 
coefficient  is  P  +  Q,  and  r+2  is  the  exponent  of  x  increased 
byl. 

Therefore  the  Jifth  law  holds  with  respect  to  (3). 

Hence,  if  the  laws  of  Art.  443  hold  for  any  power  of 
a  4-  X  whose  exponent  is  a  positive  integer,  they  also  hold 
for  a  power  whose  exponent  is  greater  by  1. 

But  the  laws  have  been  shown  to  hold  for  (a  +  x)'^,  and 
hence  they  also  hold  for  {a  -\-  xy ;  and  since  they  hold  for 
(a  +  xy,  they  also  hold  for  (a  +  x^;  and  so  on. 

Therefore  the  laws  hold  when  the  exponent  is  any  positive 
integer. 

By  aid  of  the  fifth  law,  the  coefficients  of  the  successive 
terms  after  the  third,  in  the  second  member  of  (2),  may  be 
readily  found ;  thus, 

(a  +  x)"  =  a"  +  na^^^x  -i — ^^ ^ a''~'-x- 

,  n(n  —  l)(n—2)    „_t   q  ,  ,,. 

— rrivs — ^-a'''^+---    (4) 

This  result  is  called  the  Binomial  Theorem. 

Note.  In  place  of  the  denominators  1-2,  1  •  2  •  3,  etc. ,  it  is  cus- 
tomary to  write  |2,  |^,  etc.  Tlie  symbol  \n,  read  ^'■factorial  ?i,"  signi- 
fies the  product  of  the  natural  numbers  from  1  to  n  inclusive. 

445.    Putting  a  =  1  in  equation  (4),  Art.  444,  we  obtain 
(1  +  xy  =  l  +  nx-\-  ^-— — '-  x-+  -^ ^ '-  of^  H 


EXAMPLES. 

446.  In  expanding  expressions  by  the  Binomial  Theorem, 
it  is  convenient  to  obtain  the  exponents  and  coefficients  of 
the  terms  by  aid  of  the  laws  of  Art.  443,  which  have  been 
proved  to  hold  for  any  positive  integral  exponent. 


312  COLLEGE   ALGEBRA. 

1.  Expand  (a  +  a:) ^ 

The  exponent  of  a  in  the  first  term  is  5,  and  decreases  by 
1  in  each  succeeding  term. 

The  exponent  of  x  in  the  second  term  is  1,  and  increases 
by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1 ;  of  the  second  term, 
5 ;  multiplying  5,  the  coefficient  of  the  second  term,  by  4, 
the  exponent  of  a  in  that  term,  and  dividing  the  result  by 
the  exponent  of  x  increased  by  1,  or  2,  we  have  10  as  the 
coefficient  of  the  third  term  ;  and  so  on.     Hence, 

(a  +  xy  =  a'  +  5a'x  +  lOaV  +  lOa'x^  +  Saa;"  +  af. 

Note  1.  The  coeflBcients  of  terms  equally  distant  from  the  begin- 
ning and  end  of  the  expansion  are  equal.  Thus  the  coefficients  of  the 
latter  half  of  an  expansion  may  be  written  out  from  the  first  half. 

2.  .Expand  (1  +  2 a;'')''. 
(l  +  2a;'')«^  =  [l+(2x^)]« 

=  1«+    6 -1^.(20;^)  +15-l*-(2x^)2+20-13.(2a;^)'^ 

+  15.12.(2x0^-1-  6-1  ■{2xy  +  (2xy 
=  1  +  12  x^  +  GO  or^  +  160  x''  +  240  x""  +  192  xr'> 

Note  2.  If  the  first  term  of  the  binomial  is  a  number  expressed  in 
Arabic  numerals,  it  is  convenient  to  write  the  exponents  at  first  with- 
out reduction.  The  result  should  afterward^  be  reduced  to  its  simplest 
form. 

Note  3.  If  either  term  of  the  binomial  has  a  coefficient  or  exponent 
other  than  unity,  it  should  be  enclosed  in  a  parenthesis  before  apply- 
ing the  laws. 

3.  Expand  (Sm'^  -  Vny. 

=  [(3m-^)  +  (-nO]* 

=  (;]m"^)*  +  4(3m-2)^'(_„')  +  0(3m"^)-(-w^)- 

+  4(3m-^)  (-n^)^  +  (-7t^)* 
=  81  ?u  -  —  1 08  vr  '^  ?(^  +54  wi  '  n''  —  12  m~  -  n  +  n^. 


THE   BINOMIAL   THEOREM.  313 

Note  4.  If  the  second  term  of  the  binomial  is  negative,  it  should 
be  enclosed,  negative  sign  and  all,  in  a  parenthesis  before  applying  the 
laws.  In  reducing  afterwards,  care  must  be  taken  to  apply  the  prin- 
ciples of  Art.  109. 


Expand  tlie  following 
4.    {a  +  xf. 


3\5 

5.    {a-xy.  -   13.    (^a^h-'+a-"-h")\ 


6.    (1 +»;)'•  14.    (Va^^  +  4v''a)*. 

-7.  (ct+d-r-  .  15.  ri^-^Y- 

8.  (m-*-n-)«.  16.    U'^-lx^. 

9.  (x--2^2'')^  -^     17.    {x'^  +  ^y-'y. 

10.    (a«  +  5V^)'.  18.    (3a-W&-6"^^a)'- 

A  trinomial  may  be  raised  to  any  power  by  the  Binomial 
Theorem  if  two  of  its  terms  are  enclosed  in  a  parenthesis 
and  regarded  as  a  single  term. 

20.    Expand  (ar-.2a;-2)l 
{x'-2x-2y 

=  [(x2-2x)  +  (-2)y 

=  {x^  -2xy  +  4(x2  -2xy{-  2)  +  6(x-^  -  2xy{-2y 

+  4.{x^-2x){-2y  +  {-2y 
=  a;8  _  8  xJ  +  24  .r«  -  32  x'  + 1 G  .x-^  -  8  (a;«  -  6  .r^  +  1 2  .i;^  -  8  x"') 

+  24  (a;^  -  4  x""  +  4  .i;-)  -  32  (x-  -  2  x)  +  IG 
=  a;» -  8  .t'  +  IC  x'  +  IG  :c-'  -  5G  a;'  -  32  .r'  +  G4  x'  +  C4  x  +  1 G. 


314  COLLEGE   ALGEBRA. 

Expand  the  following  : 

21.'  (x^  +  x  +  iy.  24.  {2x'  +  x  +  3y. 

22.  {3a--ax-4.x'y.  25.  (l-x  +  x'y. 

23.  {l+2x-c^y.  26.  (x'  +  2x-2y. 

447.    To  find  the  rtli  or  general  term  in  the  exjyansion  oj 
(a  +  a;)". 

The  following  laws  will  be  observed  to  hold  for  any  term 
in  the  expansion  of  (a  +  x)",  in  equation  (4),  Art.  444 : 

1.  The  exponent  of  x  is  less  by  1  than  the  number  of  the 
term. 

2.  The  exponent  of  a  is  n  minus  the  exponent  of  x. 

3.  The  last  factor  of  the  numerator  is  greater  by  1  than 
the  exponent  of  a. 

4.  The  last  factor  of  the  denominator  is  the  same  as  the 
exponent  of  x. 

Therefore,  in  the  rth  term,  the  exponent  of  x  will  be  r— 1. 
The  exponent  of  a  will  be  n  —  (r  —  1),  or  n  —  ?•  -|-  1. 
The  last  factor  of  the  numerator  will  be  n  —  r-\  2. 
The  last  factor  of  the  denominator  will  be  r  —  1. 

Hence,  the  rth  term 

^  71  (n  -  1)  (n  -  2)  ■  ■  ■  (n  -  r  +  2)  ^^„_.^j  ^^._, 
1.2.3."(r-l) 

EXAMPLES. 
1.    Find  the  eighth  term  of  (3a^  -  6-^)". 

In  this  case,  r  =  8,  and  ti  =  11 ;  hence  the  eighth  term 
^11. 10.9.8. 7. 6-5        X  . 

1.2.3.4.5.6.7   ^       ^  ^  ' 

=  330 .  (81  a-)  (  -  ?>  ')  =  -  26730  d'b'. 


THE   BINOMIAL   THEOREM.  315 

Note.   Notes  3  and  4,  Art.  446,  apply  with  equal  force  to  the 
examples  in  the  present  article. 

Find  the 

2.  Seventh  term  of  (a  -|-  xy\ 

3.  Sixth  term  of  (1  +  m)'«. 

4.  Eighth  term  of  (c  -  ciy-. 
—       5.    Fifth  terra  of  (1  -  a-y^ 

6.  Seventh  term  of  /'-+- 

\b     a 

7.  Tenth  term  of  (x™  -  V^)". 

8.  Seventh  term  of  (  a^^  —  ~ab 


2 
9.    Eighth  term  of  (x-'  +  2iJy^ 

10.    Sixth  term  of  (a"  ^-  -  3x^y\ 

9     \  13 

V«i  +  -^^ )  • 

-\/mJ 


12.    Find  the  middle  term 


-e-r 


13.    Find  the  term  involving  x^^  in  far  +  — ^ 


14.    Find  the  term  involving  x"  in  (  2ar= =- ) '. 


816 


COLLEGE   ALGEBRA. 


XXXI.     CONVERGENCY  AND  DIVERGENCY 
OF  SERIES. 

448.  A  Finite  Series  is  a  series  having  a  finite  number 
of  terms. 

An  Infinite  Series  is  a  series  the  number  of  whose  terms 
is  unlimited. 

The  progressions,  in  general,  are  examples  of  finite  series; 
but  in  Art.  429  we  considered  infinite  geometrical  series. 

449.  Infinite  series  may  be  developed  by  the  processes  of 
Division  and  Evolution. 

Let  it  be  required,  for  example,  to  divide  1  by  1  —  x. 

l-x)l{l-\-x-^x'  +  •■• 
1-x 

X 

X  —  a? 


The  quotient  is  obtained  in  the  form  of  the  infinite  series 

l-\-x  +  x?-\ 

Again,  let  it  be  required  to  find  the  square  root  of  1  +  a;. 


l+x\l+-^--^  + 


^       o 

1 

2  +  1 

X 

XT 

^  +  ^ 

4 

~^' 

ar' 

2  -{-x- 

~  8 

4 

1  + 


The  result  is  obtained  in  the  form  of  the  infinite  series 

X        0"    , 


CONVEKGENCY  AND  DIVERGENCY  OF  SERIES.    317 

Infinite  series  may  also  be  developed  by  other  methods, 
one  of  the  most  important  of  which  will  be  considered  in 
Chapter  XXXII. 

450.  A  series  is  said  to  be  convergent  when  the  sum  of 
the  first  n  terms  approaches  a  certain  fixed  quantity  as  a 
limit  (Art.  209),  when  n  is  indefinitely  increased;  this 
limiting  value  is  calleS.  the  Sum  of  the  Series. 

A  series  is  also  said  to  be  convergent  when  the  sum  of  all 
its  terms  is  equal  to  a  fixed  finite  quantity. 

A  series  is  said  to  be  divergent  when  the  sum  of  the  first 
n  terms  can  be  made  to  numerically  exceed  any  assigned 
quantity,  however  great,  by  taking  n  sufficiently  great. 

451.  Consider,  for  example,  the  infinite  series 

l+x  +  x-  +  x^-\ 

I.  Suppose  X  =  Xi,  where  Xi  is  numerically  <  1. 
The  sum  of  the  first  n  terms  is  now 

1  +  a^i  +  a;,-  +  •••  +  a;i"-^  =  ^  ~  ^'"  (Art.  115). 

When  n  is  indefinitely  increased,  x^'  decreases  indefi- 
nitely in  absolute  value,  and  approaches  the  limit  0. 

1  —  a;"  1 

Therefore  '-^  approaches  the  limit • 

1  —  Xi  1  —  Xi 

That  is,  the  sum  of  the  first  n  terms  approaches  a  certain 
fixed  quantity  as  a  limit,  when  n  is  indefinitely  increased. 
Hence  the  series  is  convergent  when  x  is  numerically  <  1, 

II.  Suppose  X  =  1. 

In  this  case,  each  term  of  the  series  is  equal  to  1,  and  the 
sum  of  the  first  n  terms  is  equal  to  n;  and  this  sum  can  be 
made  to  exceed  any  assigned  quantity,  however  great,  by 
taking  n  sufficiently  great. 

Hence  the  series  is  divergent  when  x  =  1. 


318  COLLEGE   ALGEBRA. 

III.  Suppose  a;  =  — 1. 

In  this  case  the  series  takes  the  form 

1-1  +  1-1  +  ...; 

and  the  sum  of  the  lirst  n  terms  is  either  1  or  0  according 
as  n  is  odd  or  even. 

Hence  the  series  is  neither  convergent  nor  divergent  when 
X-  =  -  1. 

IV.  Suppose  x=  Xi,  wlaere  Xi  is  numerically  >  1. 
The  sum  of  the  first  n  terms  is 

1  +  Xi  +  xi^  +  ...  +  xr'  =  "^LJzl   (Art.  115). 

X'l  —  1 

x"  —  1 

By  taking  n  sufficiently  great,  the  expression  — —  can 

Xi  —  1    • 

be  made  to  numerically  exceed  any  assigned  quantity,  how- 
ever great. 

Hence  the  series  is  divergent  when  x  is  numerically  >  1. 

452.    Consider  the  infinite  series 

1  +  x  +  .^•^  +  a.-^  H , 

developed  by  the  fraction (Art.  449). 

Let  x  =  .l,  in  which  case  the  series  is  convergent  (Art.  451). 
The  series  now  takes  the  form  1  +  .1  +  .01  +  .001  +  ..., 

while  the  value  of  the  fraction  is  — ,  or  -— . 

.9  9 

In  this  case,  however  great  the  number  of  terms  taken, 

their  sum  will  never  exactly  equal  — ;  but  it  approaches 
this  value  as  a  limit  (Art.  430). 

Thus,  if  an  infinite  series  is  convergent,  the  greater  the 
number  of  terms  taken,  the  more  nearly,  does  their  sum 
approach  to  the  value  of  the  expression  from  which  the 
series  was  developed;  and  the  sum  of  the  series  (Art.  450) 
is  (Hiua.l  to  tlic  value  of  this  expression. 


CONVERGENCY  AND  DIVERGENCY  OF  SERIES.     819 

Again,  let  x  —  10,  in  which  case  the  series  is  divergent. 

The  series  now  takes  the  form  1  +  10  +  100  +  1000  +  •••, 

while  the  value  of  the  fraction  is  ,  or 

1-10  9 

In  this  case  it  is  evident  that,  the  greater  the  number  of 

terms  taken,  the   more  does  their    sum  diverge  from  the 

value 

9 

Thus,  if  an  infinite  series  is  divergent,  the  greater  the 

number  of  terms  taken,  the  more  does  their  sum  diverge 

from  the  value  of  the  expression  from  which  the  series  was 

developed. 

It  follows  from  the  above  that  an  injinite  series  cannot  be 
used  for  the  inirposes  of  demonstration,  unless  it  is  convergent. 

Note.  It  will  be  understood  hereafter  that,  in  every  expression  in- 
volving a  convergent  infinite  series,  the  sum  of  the  series  is  meant. 

For  example,  the  product  of  two  convergent  infinite  series  will  be 
understood  as  signifying  the  product  of  their  sums. 


ELEMENTARY  THEOREMS   ON  THE   CONVERGENCY 
AND     DIVERGENCY   OF   SERIES. 

453.  The  infinite  series 

a  +  hx  +  cxr  +  dx"'  +  ••• 

is  convergent  when  .'^  =  0 ;  for  the  sum  of  all  the  terms  is 
equal  to  a  when  a;  =  0. 

454.  If  all  the  terms  of  an  infinite  series  are  of  the  same 
sign,  the  series  must  he  either  convergent  or  divergent. 

Note.  An  infinite  series  may  be  neither  convergent  nor  divergent; 
an  example  of  this  has  been  given  in  Art.  4-51,  III. 
Such  series  are  called  indeterminate  or  neutral. 

In  this  case,  the  greater  the  number  of  terms  taken,  the 
greater  will  be  the  absolute  value  of  their  sum. 


320  COLLEGE   ALGEBRA. 

Therefore,  either  the  sum  of  the  first  n  terms  approaches 
a  certain  fixed  quantity  as  a  limit,  when  n  is  indefinitely 
increased ;  or  else  the  sum  of  the  first  n  terms  can  be  made 
to  numerically  exceed  any  assigned  quantity,  however  great, 
by  taking  n  sufficiently  great. 

Hence  the  series  must  be  either  convergent  or  divergent. 

455.  By  Art.  454,  if  all  the  terms  of  an  infinite  series  are 
of  the  same  sign,  the  series  must  be  either  convergent  or 
divergent ;  hence, 

I.  If  the  sum  of  the  first  n  terms  is  numerically  less  than 
a  certain  fixed  quantity,  however  great  n  may  be,  the  series 
is  convergent. 

II.  If  each  term  of  the  series  is  numerically  less  than 
the  corresponding  term  of  another  infinite  series  whose 
terms  are  all  of  the  same  sign,  and  which  is  known  to  be 
convergent,  the  first  series  is  convergent. 

III.  If  each  term  of  the  series  is  numerically  greater 
than  the  corresponding  term  of  another  infinite  series 
whose  terms  are  all  of  the  same  sign,  and  which  is  known 
to  be  divergent,  the  first  series  is  divergent. 

456.  If  all  the  terms  of  an  infinite  series  are  of  the  same 
sign,  and  each  term  is  numerically  greater  than  some  assigned 
finite  quantity,  however  small,  the  series  is  divergerit. 

For  if  each  term  of  the  series  is  numerically  greater  than 
a,  the  sum  of  the  first  n  terms  is  numerically  greater  than 
na,  and  hence  can  be  made  to  numerically  exceed  any 
assigned  quantity,  however  great,  by  taking  w  sufficiently 
great. 

Therefore  the  series  is  divergent. 

457.  It  follows  from  Art.  456  that,  i^H;l7he  terms  of  an 
^infinite  series  are  of  the  same  sign,  the  series  is  divergent 

unless  its  7ith  term  approaches  the  limit  0  when  n  is  indefi- 
nitely increased. 


CONVERGENCY  AND  DIVERGENCY  OF  SERIES.     321 

458.  If  the  terms  of  an  ivjinite  series  are  alternately  iiositive 
and  negative,  and  each  term,  is  numerically  less  than  the  lire- 
ceding  term,ihe  series  is  convergent. 

For  let  tlie  series  be  Ui  —  ti^  +  tu  —  Ui+  •••  • 

It  may  be  written  in.  either  of  tlie  forms 

{u^  -  Ma)  +  (t(3  -  W4)  +  {ih  -  Ue)  +  •",  (1 ) 

or,  «i  —  (W2  —  W3)  -  (M4  —  7/5) (2) 

By  hypothesis,  each  of  the  expressions  u^  —  Uo,  u^  —  ■?'« 
etc.,  is  positive ;  that  is,  all  the  terms  of  series  (i)  are  pos- 
itive ;  and  it  is  evident  from  (2)  that  the  sum  of  the  first  n 
terms  of  the  series  is  less  than  ii^,  however  great  n  may  be. 

Therefore,  by  Art.  455,  I.,  the  series  is  convergent. 

As  an  example  of  the  above,  the  infinite  series 


1-1+1-1+ 
2      3      4 


is  convergent. 


459.  If  all  the  terms  of  an  infinite  series  are  of  the  same 
sign,  and  the  ratio  of  each  term  to  the  preceding  term  is  less 
than  a  certain  quantity  which  is  itself  less  than  unity,  the  series 
is  convergent. 

Let  the  series  be 

%  +  «2  +  «3-l hw,_i  +  ?(^H ;  (1) 

and  suppose  ~<x,  ^  <  a;,  . . .,  ^  <x,  .. .,  where  a;  is  <  1. 

Ui  U2  K-r-l 

By  Art.  451,  I.,  since  a;  is  <  1,  the  infinite  series 

Mi(l  +  a;  +  x2  +  x'^+...)  (2) 

is  convergent. 

Multiplying  together  the  first  r —1  of  the  given  inequali- 
ties (Art.  226),  we  have 

— ^-^^ —  <  af  ~^ ;  whence,  —  <  af  ~^ ;  or,  u^  <  ?<,af  '. 


322  COLLEGE    ALGEBRA. 

That  is,  the  rth.term  of  (1)  is  numerically  less  than  the 
j*th  term  of  series  (2),  which  is  known  to  be  convergent. 
Therefore,  by  Art.  455,  II.,  the  series  (1)  is  convergent. 

460.  If  all  the  terms  of  an  infinite  series  are  of  the  same 
sign,  and  the  ratio  of  each  term  to  the  preceding  term  is  either 
equal  to  unity,  or  greater  than  unity,  the  series  is  divergent. 

Let  the  series  be  xi^  +  u^.  +  «3  +  ^<4  +  — 

If  the  ratio  is  equal  to  unity,  each  term  of  the  series  is 
equal  to  w^;  and  if  the  ratio  is  greater  than  unity,  each 
term,  after  the  first,  is  numerically  greater  than  Wj. 

In  either  case  the  series  is  divergent  by  Art.  457. 

461.  If  an  infinite  series,  in  which  all  the  terms  are  of 
the  same  sign,  is  convergent,  it  will  remain  so  after  the 
signs  of  any  number  of  terms  are  changed ;  for  the  sum  of 
the  first  n  terms,  when  n  is  indefinitely  increased,  will  be 
numerically  less  in  the  latter  case  than  in  the  former. 

It  follows  from  the  above  that  the  theorems  of  Arts.  455, 
II.,  and  459  hold  when  the  terms  are  not  all  of  the  same 
sign. 

462.  If  an  infinite  series  is  convergent  or  divergent,  it 
will  evidently  remain  convergent  or  divergent  after  any 
finite  number  of  terms  have  been  added  to,  or  subtracted 
from  it. 

Therefore,  in  testing  a  series  for  convergency  or  diver- 
gency, ^«e  may  commey\ce  at  any  assigned  term,  taking  no 
account  of  the  preceding  terms. 

Thus,  the  theorem  of  Art.  459  might  be  stated  as  follows  : 

"  If,  after  any  assigned  term,  the  terms  of  an  infinite  series 
are  all  of  the  same  sign,  and  the  ratio  of  each  term  to  the 
preceding  term  is  less  than  a  certain  quantity  which  is 
itself  less  than  unity,  the  series  is  convergent." 

Similar  considerations  hold  with  respect  to  Arts.  455  to 
458  inclusive,  and  Art.  4G0. 


CONVERGENCy  AND  DIVERGENCY  OF  SERIES.     823 

^    463.    The  infinite  series 

a  +  &a;  +  ca^  +  daf'  -1-  •  •  •  +  A;a;''~^  +  lx'  -\-  ••• 
is  convergent  when  x  is  taken  sufficiently  smalL 


For  the  ratio  of  the  (r+l)st  term  to  the  vth  term 


Ix 
or  — ;  and  whatever  the  values  of  I  and  A:,  x  may  be  taken  so 

^  Ix 

small  that  the  ratio  —  shall  always  be  numerically  less  than 

/c 
q,  where  g  is  a  quantity  numerically  less  than  1. 

Therefore  the  series  is  convergent  by  Art.  459. 

464.  To  prove  that  the  bt finite  series 

1+^+1  +  1+-+-+- 
2       3  r 

is  convergent  ivhen  x  is  numerically  <  1. 

The  ratio  of  the  (r  +  l)st  term  to  the  ^-th  terra 

=  ^  ^  •^'"^'   =^X  ^'~^  =  x{r-l)  ^  ^A  _  1 
r       r  —  1       r        x''^  r  \        '" 

Since  1 is  less  than  1,  the  ratio  of  the  (?•  +  l)st  term 

to  the  rth  term  is  numerically  less  than  x\  that  is,  it  is 
numerically  less  than  a  quantity  which  is  itself  numerically 
less  than  1. 

Therefore,  by  Art.  459,  the  series  is  convergent. 

465.  To  prove  that  the  infinite  series 

is  convergent  for  every  value  ofx. 

The  ratio  of  the  (r+l)st  term  to  the  rth  term 

_  a;''  .     x'-'^   _  ^'  _     ^     y  ^  ~  1  _     ^     f-i      1 
[r      [r  —  1      r      r  —  1         r         r  —  1 


324  COLLEGE   ALGEBRA. 

If  r  is  so  taken  that  ?-  —  1  is  numerically  greater  than  x, 

will  be  numerically  less  than  1 ;  and  as  r  increases 

r  —  1 

beyond  this  value,  — '—-  decreases  indefinitely  in  absolute 

value.  ~ 

Also,  1 is  less  than  1. 

r 

Therefore,  commencing  at  a  certain  assigned  term,  the  ratio 
of  each  term  to  the  preceding  term  is  numerically  less  than 

a  quantity ,  which  is  itself  numerically  less  than  1. 

Hence  the  series  is  convergent  by  Art.  462. 

466.    To  irrove  that  the  infinite  series  (compare  Art.  445) 

\2  [3  ' 

where  n  is  any  real  number,  not  a  ^wsitive  integer,  is  convergent 
when  X  is  numerically  <  1. 

By  Art.  447,  the  ratio  of  the  (r  +  l)st  term  to  the  rth 
term 
^  n(n-l)-(n-r  +  l)  ^r  _  n(n  -  1) -.  (n  -  r  +  2)  ^,_, 


\r-l 


^  n{n  -!)•••  {n  -r  +  2){n  ^'r  +  1) ^, 
1.2.3.-.(r-l)r 

^^  l-2-3-(r-l) 


'n{n  —  1)  •••  {n  —  r  -\-2)x''~^ 


^-^•  +  l.._A«+l_n^. 


■X 

r 

I.    Suppose  n  positive. 

?i  -I-  1 
If  r  is  taken,  >  n,      ^     is  positive  and  <  2 ;  and  hence 

Vl^L±  _  1  is  numerically  <  1.        ,  .\t-f 


V 


CONVERGENCY  AND  DIVERGENCY  OF  SERIES.     325 

II.    Suppose  n  negative,  and  numerically  <  1. 

11  A-  1 
Then,  whatever  the  value  of  r,  is  positive  and  <  1 ; 

4-1  ^ 

and  hence  ^^^^t 1  is  numerically  <  1. 

r 

ni.    Suppose  n  negative,  and  numerically  >  1 ;  and  let 
w  =  —  n',  where  n'  is  positive  and  >  1. 

Then,       n-r  +  l^^-n'-r  +  l^^i;^-l 
r  r  —  1  r 


"■  +iV-fi-i)- 


1 


If  r  is  taken  sufficiently  great,  — [- 1  may  be  made  to 

r  —  1 

differ  from  1  by  less  than  any  assigned  quantity,  however 

small. 

Hence,  if  r  is  taken  sufficiently  great,  f— hi  )a;  may 

be  made  numerically  less  than  1,  since  x  is  numerically  less 

than  1;  and  as  r  increases  beyond  this  value,  (— |-l)a; 

decreases  indefinitely  in  absolute  value.  v  ~  J 

Also,  1 is  less  than  1. 

r 

Thus,  in  the  first  and  third  cases  after  a  certain  assigned 
term,  and  in  the  second  case  always,  the  ratio  of  the  (?•+!) st 
term  to  the  rth  term  is  numerically  less  than  a  quantity 
which  is  itself  numerically  less  than  1. 

Therefore  by  Art.  462,  the  series  is  convergent  in  each  of 
the  above  cases. 

IV.    If  n  =  —  1,  the  series  takes  the  form 
1  —  a;  +  x- —  x^  +  •••, 
which  is  convergent  by  Art.  458. 


32G  COLLEGE   ALGEBRA. 

467.    To  irrove  that  the  infinite  series 

l  +  |  +  |  +  j;  +  -  (1) 

is  convergent  tvhen  n  is  >  1,  and  divergent  lohen  ?i  =  1  orn  <  1. 

I.  If  n  is  >  1,  the  second  and  third  terms  are  together 

119  4 

<  —  +  — ,  or  <  -^^ ;    the  next  four  terms  are  together  <  — ; 

8 
the  next  eight  are  together  <  — ;  and  so  on. 

8" 

Therefore  the  series  is  less  than  the  series 

which  is  known  to  be  convergent  by  Art.  451-,  I. 

Hence  tlie  given  series  is  convergent  (Art.  455,  II.). 

II.  If  n  =  1,  the  series  becomes 

i+l+i+h <2) 

2  1 

The  third  and  fourth  terms  are  together  >  -,  or  >  -  •, 

the  next  four  terms  are  together  >  -,  or  >  - ;  and  so  on. 

8  2 

Hence,  by  taking  a  sufficiently  great  number  of  terms^ 
their  sum  may  be  made  to  exceed  any  assigned  quantity, 
however  great. 

Therefore  the  series  is  divergent. 

III.  If  n  is  positive  and  <1,  or  negative,  each  term  of 
(1)  is  greater  than  the  corresponding  term  of  series  (2). 

Hence  the  given  series  is  divergent  (Art.  455,  III.). 


CONVERGENCY  AND  DIVERGENCY  OF  SERIES.      327 

EXAMPLES. 
468.   Expand  each  of  the  following  to  four  terms  : 
.     .     2-5aj  a.    2-5.T  +  6a;'^  „        ,-— — 


1  +  ar 


1  +  4  a;  —  o  a;- 


1 

2-5.T  +  6a;'^ 

3_7x-2_x-^ 

5. 

Vl-2a^ 

8.    Vl  +  ar^. 


-.  _     6.    Va--4a&  +  6l        9.    -^8a^-'6b. 


10.    Prove  that  the  infinite  series 

1  +  1+1+1  +  ...., 
is  convergent. 

The  series  is  less  than  the  series 

14.1,14.1-L 

which  is  known  to  be  convergent  by  Art.  451,  I. 
Therefore  the  series  is  convergent  (Art.  455,  II.). 

Examine  whether  the  following  infinite  series  are  con- 
vergent or  divergent :  - 

il     (3     11^         '  1.2^2-3    3.4^4-5 

13.   l+|  +  |  +  ^.  +  --      16.   l+f3  +  |  +  J  +  --       '+ 
"•   l  +  r2  +  2^  +  3-^  +  4-^5+--  . 


328  COLLEGE   ALGEBRA. 


XXXII.     THE   THEOREM    OF    UNDETER- 
MINED    COEFFICIENTS. 

469.  An  important  method  for  expanding  expressions 
into  series  is  based  on  the  following  theorem,  known  as  the 
Theorem  of  Undetermined  Coefficients : 

If  the  series  A  +  Bx  +Cx^-\-Daf+  •••  is  always  equal  to  the 
series  A'+  B'x+C'u:^+  D'af+  •••,  when  x  has  any  value  which 
makes  both  series  convergent,  the  coefficients  of  like  powers  of  x 
in  the  two  series  will  he  equal;  that  is,  A=A',  B  =  B',  C  =  C', 
etc. 

For  since  the  equation 

A  +  Bx  +  Car  +  Dx^  +  ■••  =  A'+  B'x+C'x'  +  D'x^  +  ... 

is  satisfied  when  x  has  any  value  which  makes  both  mem- 
bers convergent,  and  since  both  members  are  convergent 
when  x  =  0  (Art.  453),  it  follows  that  the  equation  is  satis- 
fied when  x  =  0. 

Putting  a;  =  0,  we  have  A  =  A'. 

Subtracting  A  from  the  first  member  of  the  equation,  and 
its  equal  A'  from  the  second  member,  we  obtain 

Bx+Cx"  +  Z>x-3  +  ...  =  B'x+C'x-  +  D'x"  -\ 

Dividing  through  by  x, 

B  +Cx  +  Dx"  +  -•■  =  B'  +  C'x  +  D'x"  -i 

This  equation  also  is  satisfied  when  x  has  any  value  wliich 
makes  both  members  convergent ;  and  putting  x=0,  we  have 

B=B'. 

In  like  manner,  we  may  prove  C  =  C',  D  =  D',  etc. 

Note.  The  above  demonstration  is  the  one  usually  giveti  in  text- 
books on  Algebra;  it  is,  however,  open  to  olijection  in  one  respect. 


UNDETERMINED   COEFFICIENTS.  329 

It  is  demonstrated  rigorously  that 

5x  +  Cic^  +  2>x3  +  ...  =  B'x  +  (7'x2  +  D'oc^  +  -, 
or,  x{B+CxfDx''+-)  =  x{B<+C'x-\-D'x^+-),  (1) 

when  X  has  any  value  which  makes  both  members  convergent,  including 
the  value  zero. 

But  when  we  divide  through  by  x,  and  put  the  result  m  the  form 

5  +  Cx  +  Dx^  +•••  =  £'+  C'x  +  D'x:^  +  •••,  (2) 

all  that  we  know  about  this  equation  is,  that  it  is  satisfied  by  every 
value  of  X,  except  zero.,  which  makes  both  members  convergent. 

We  cannot  assert  that  the  equation  is  satisfied  when  x  =  0 ;  for 

equation  (1)  is  satisfied  when  x  =  0,  even  if  B+  Cx-\-  Dx:^-\ and 

B'  +  C'x  +  D'x^  +  •••  are  unequal  for  this  value  of  *. 

In  order  to  demiinstrate  rigorously  that  B  =  B\  we  may  proceed  as 
follows : 

Both  members  of  (2)  are  convergent,  and  the  equation  therefore 
satisfied,  when  x  is  taken  sufficiently  small  (Art.  463),  and  for  all 
smaller  values  of  x,  except  zero. 

But  if  X  is  taken  sufficiently  small,  the  first  member  may  be  made 
to  differ  from  5,  and  the  second  member  from  J3',  by  less  than  any 
assigned  quantity,  however  small. 

■      Hence  B  and  B'  cannot  differ  by  any  assigned  quantity,  however 
small,  and  are  therefore  equal. 

470.  A  finite  series  being  always  convergent,  it  follows 
from  the  preceding  article  that  if  two  finite  series 

A  +  Bx+Cx--\ +  Ax"  and  A'  +  B'x  +  C'x-  -f  •  •  •  +  K'x"-, 

are  equal  for  every  value  of  x,  the  coefficients  of  like  powers 
of  X  in  the  two  series  are  equal. 

-    ,  EXPANSION  OF  FRACTIONS   INTO   SERIES. 

9  _  3  3j2  ^.3 

471.  lo  Expand  -^ '—,  in  ascending  powers  of  x. 

1  —  2x  +  3a^ 

We  have  seen  in  Art.  449  that  a  fraction  of  the  above  form 
may  be  expanded  into  a  series  by  dividing  the  numerator 
by  the  denominator ;  we  therefore  know  that  the  proposed 
expansion  is  possible. 


330  COLLEGE   ALGEBRA. 


Assume  then 


A  +  Bx-{-Ca^  +  Dx'  +  Ex*  +  —  ;      (1) 


1  — 2a;H-3ic^ 

where  A,  B,  C,  D,  E,  ...,  are  quantities  independent  of  x. 

Clearing  of  fractions,  and  collecting  the  terms  in  the  second 
member  involving  like  powers  of  x,  we  have 

»*+....  (2) 


Bx+    C 

ar'+    D 

.T^+     E 

2A     -2B 

-20 

-2D 

+  3.4 

+  35 

+  3C 

The  second  member  of  (1)  must  express  the  value  of  the 
fraction  for  every  value  of  x  which  makes  the  series  con- 
vergent (Art.  452). 

Hence  equation  (2)  is  satisfied  when  x  has  any  value 
which  makes  both  members  convergent,  and  by  the  Theorem 
of  Undetermined  Coefficients,  the  coefficients  of  like  powers 
of  X  in  the  two  series  are  equal ;  that  is, 

A=     2. 
B-2A=     0;  whence,  5  =  2^  =4. 

C-2B  +  3A  =  -3;  whence,  C  =  2B-3A-3=-l. 
lj_2C-\-3B  =  -l;  whence,  D  =  2C-3B-1= -15. 
E-2D+3C  =     0;  whence,  E=2D-3C       =-27;  etc. 

Substituting  these  values  in  (1),  we  have 

l^zl^l^l^  =  2  +  4:x-x''-15x'-27x' 

l-2a;  +  3ar 

Tlie  result  may  be  verified  by  division. 

Note  1.  A  vertical  lino,  called  a  bar,  is  often  used  in  place  of  a 
parenthesis ;  thus, 

+     Ji\x  \s  equivalent  to  (B  —  2A)  x. 

-2a\ 

Note  2.  The  result  expresses  the  value  of  the  given  fraction  only 
for  such  values  of  x  as  make  the  series  convergent  (Art.  452). 


UNDETERMINED   COEFFICIENTS.  331 

If  the  numerator  and  denominator  contain  only  even 
powers  of  x,  the  operation  may  be  abridged  by  assuming  a 
series  containing  only  the  even  powers  of  x. 

Thus,  if  the  fraction  were  ^   ^^   ^ — — -,  we  should  assume 
1  — 3ar+5a;"* 

it  equal  to  ^  +  B^^  +  Cx*  -|-  Dx^  +  Ex^  -\ 

In  like  manner,  if  the  numerator  contains  only  odd  powers 
of  X,  and  the  denominator  only  even  powers,  we  should 
assume  a  series  containing  only  the  odd  powers  of  x. 

If  every  term  of  the  numerator  contains  x,  we  may  assume 
a  series  commencing  with  the  lowest  power  of  x  in  the  nu- 
merator. 

EXAMPLES. 

Expand  each  of  the  following  to  five  terms,  in  ascending 
powers  of  x : 

2     ^-^  6     ^-^-^^  10     2-3a;  +  4a.-^ 

'    1  +  x  '    ^  ■-■  •    -       ^ 


3     -  +  5a; 
l-'dx 


4.   ^-^^\  8.   ^-■■^^■•^ .  12. 


l  +  bx" 

2x 


1+x  +  x^ 

X 

-Sx^-x" 

1 

-2x-x- 

2 

—  x  +  x^ 

1-x' 

l-2ar^ 

11. 


13. 


l+2x-5x^ 

a;^  +  2ar^ 

2-x-x'' 

3  +  a!-2a;3 

3  -  a^  +  a;3 

l-3a)2 

3-2ar^  I  +  2a;-3a,-2  2-3i»-2ar^ 

If  the  lowest  poAver  of  x  in  the  denominator  is  higher  than 
the  lowest  power  in  the  numerator,  we  may  determine  by 
actual  division  what  power  of  a;  will  occur  in  the  first  term 
of  the  expansion ;  we  should  then  assume  the  fraction  equal 
to  a  series  commencing  with  this  power  of  x,  the  exponents 
of  X  in  the  succeeding  terms  increasing  by  unity  as  before. 

14.    Expand  — - — — -  in  ascending  powers  of  x. 
3ar  —or 

Dividing  1  by  3x^,  the  quotient  is  ^- 


33i 


COLLEGE   ALGEBRA. 


We  then  assume, 

. — =  Ax-'  +  Bx-''+C+Dx  +  Ea^  +  --       (3) 

Clearing  of  fractions, 

l  =  3^  +  35|a;  +  3C|a;-  +  3i)ja;'  +  3^|a;*+---. 

-  a\  -  b\    -  c\    -  d\ 

Equating  the  coefficients  of  like  powers  of  x, 
3A  =  1 
3B-A  =  0: 
3C-B=0 
3D- C=0 
3E-D  =  0;  etc. 

Whence,  ^  =  |  i?  =  |   C'=l.,  D  =  l^,  E  =  ^^,  etc. 
Substituting  in  (3),  we  have 

_J— —  ^"-4-  — -I-— -l-^  +  — +  •  •  • 
Sx'-x^"   3  "^   9  ^27  "^81^243 

Expand  to  five  terms  in  ascending  powers  of  x 


15. 

3a;--4ar^ 

16. 

1+x-o^ 

x-2x^-\-3x' 

17. 
18. 


l-2a.-^-.r\ 

X^  +  X^  —  X* 

3-2a;  +  .^•^ 
2x'-x*-'2x'^ 


EXPANSION   OF   SURDS  INTO   SERIES. 

472.    1.  Expand  Vl  —  x  in  ascending  powers  of  x. 

We  have  seen  in  Art.  449  that  the  square  root  of  an  imper- 
fect square  can  be  expanded  into  a  series  by  the  process  of 
Evolution ;  we  therefore  know  that  the  proposed  expansion 
is  possible.     Assume  then 

Vr^^  =  ^  +  Bx  +  Cx'  +  D.r  +  E,i^  +  ■■•  -         (1) 


UNDETERMINED   COEFFICIENTS. 


833 


Squcaring  both  members,  we  have  by  Art.  230, 


l-x  =  A' 


+  2AB 


x-^      B' 

x' 

x'+       C- 

+  2  AC 

+  2  AD 

+  2AE 

-{-2BC 

+  2BD 

0)'  + 


Equating  the  coefficients  of  like  powers  of  x, 
A^  =      1;  whence, -4=1. 

2  A         2 

^  =  _L 

2  A         8° 

BC^_\^ 
A  16" 


2AB  =  —  1 ;  whence,  5=  — 

J5^  -f  2 ^C  =      0 ;  whence,  (7=  - 

2AD^2BC=      0;  whence,Z)=- 

C-^2AE+2BB=      0;  whence,  £=  — 


2A 


128' 


etc. 


Substituting  these  vakies  in  (1),  we  have 


VT^^'  =  1 


X       X-        X'        ox^ 


2      8      16      128 
The  answer  may  be  verified  by  the  method  of  Art.  449. 

Note  1.   The  result  expresses  the  value  of  the  given  surd  only  for 
such  values  of  ic  as  make  the  series  convergent. 

Note  2.    The  equation  A^  =  1  gives  ^  =  ±  1  ;  and  taking  the  nega- 
tive value  of  ^4,  we  should  find  B  =  -,   C=  -,  -D  =  — ,  etc. 
2  8  15 

Thus  another  answer  to  the  example  is  —  1  +  ^  +  ^  +  —  +  •••• 

2        o        lu 

EXAMPLES, 
Expand  each  of  the  following  to  five  terms,  in  ascending 
powers  of  x : 


2.    Vl-f-2cK.         4.    VI  -  2a;  +  ax''.         6.    </l  -  x. 


3.    Vl 


b.    Vl  +x  —  x'i 


7.    Vl  +  x-\-x\ 


334  COLLEGE   ALGEBRA. 


PARTLAL   FRACTIONS. 


473.  If  the  denominator  of  a  fraction  can  be  resolved 
into  factors,  each  of  the  first  degree  in  x,  and  the  numerator 
is  of  a  lower  degree  than  the  denominator,  the  Theorem  of 
Undetermined  Coefficients  enables  us  to  express  the  given 
fraction  as  the  sum  of  two  or  more  partial  fractions,  whose 
denominators  are  factors  of  the  given  denominator,  and 
whose  numerators  are  independent  of  x. 

Case  I. 

474.  When  no  two  factors  of  the  denominator  are  equal. 

19x  + 1 

1,    Separate   — — ^ into  partial  fractions. 

Assume  ^^^+1 =  -A_  +  _JB_ 

(3ic-l)(5a;  +  2)      ^x-1      bx  +  2'  ^ 

where  A  and  B  are  quantities  independent  of  x. 

Clearing  of  fractions,  Ave  have 

19a;  +  1  =  A{ox  +  2)  +  B{^x  -  1) 

=  {5A  +  3B)x  +  2A-B.  (2) 

The  second  member  of  (1)  must  express  the  value  of  the 
given  fraction  for  every  value  of  x. 

Hence  equation  (2)  is  satisfied  by  every  value  of  x,  and 
by  Art.  470,  the  coefiicients  of  like  powers  of  x  in  the  two  ■ 
members  are  equal. 

That  is,  5^4-35  =  19, 

and  2  A-    B=    1. 

Solving  these  equations,  we  obtain  A  =  2  and  B  =  3. 

Siibstituting  in  (1),  we  havi; 

19.T  +  1  _       2  3 

•"  K, 


(3a;- 1)  (5a; +  2)      3x-l      5x  +  2 
The  result  may  be  verified  l)y  adding  the  partial  fractions 


UNDETERMINED  COEFFICIENTS.  335 

x  4-  4 

2.    Separate  — •  into  partial  fractions. 

2x  —  x~  —  x'' 

The  factors  of  2  a;  —  ar  —  ar"  are  x,  1  —  x,  and  2  -\-x  (Art. 
355).     Assume  then 

x+4       ^A ^      B      ^      C 


2x  —  X-  —  x^      X      1  —  X     2  +  cc 
Clearing  of  fractions,  we  have 

a;  +  4  =  ^(1  -  a;)  (2  +  x)  +  Bx{2  +  x)+  Cx(l  -  x). 
This  equation,  being  satisfied  by  every  value  of  x,  is  sat- 
isfied when  a;  =  0. 

Putting  a;  =  0,  we  have  4:  =  2A,ovA  —  2. 
Again,  the  equation  is  satisfied  when  x  =  1. 
Putting  a;  =  1,  we  have  5  =  3  B,  ov  B  =  — 
The  equation  is  also  satisfied  when  a;  =  —  2. 

Putting  a;  =  —  2,  we  have  2  =  —  6  C,  or  C  =  —  -• 

o 

5  _1 

rn  x  +  4:  2^3,         3 

Then,        -^, =  -  -j-  ^ h  r: 

2a;  —  ar*  —  ar*      x      1—  x      2  +  x 

2  5  1 


x      3(1 -a;)      3(2  + a;) 

Note.   The  student  should  compare  the  above  method  of  finding  A 
and  B  with  that  used  in  Example  1. 


EXAMPLES. 

Separate  each  of  the  following  into  partial  fractions  : 
3     14g;-25         g  13a; +  10  -  q     2x'-1Tx-2A 

6  a;^  —  13  x^  —  5  x 

ax  —  14  a^ 


4x2 

-25 

4x 

+  15 

3a;2 

+  5x 

x^ 

-45 

10. 


x^  —  3  ax  —  4  a^ 

7x  +  9 
2x3 -18x       -•    9  + 9a; -4x2'         ""'    4.'r-20x  +  23 


5.      "  ~T"   .       8.    — Iii±A_-.         11 


(x+l)(4x-"- 

-^J) 

2x^-20 

(x2-4)(x^'- 

■1) 

4x-14 

336  COLLEGE   ALGEBRA. 

Case  II. 

475.  WJien  all  the  factors  of  the  denominator  are  equal. 

rff 11  a;  4-  26 

Example.    Separate  ^ — •  into  partial  fractions. 

{x-dy 

If  we  attempt  to  solve  the  example  by  the  method  of 
Case  I.,  we  should  assume 

a;'' -11  a; +  26  ^     A  B  C 

{x-Sy  a;-3      x-3      x-3* 

(x  —  sy  X  —  o 

But  this  is  evidently  impossible,  for  the  given  fraction 
cannot  be  reduced  to  an  equivalent  fraction  having  x  —  3 
for  a  denominator,  and  a  numerator  independent  of  a;. 

Let  us  now  substitute  y  +  3  in  place  of  x  in  the  given 
fraction ;  we  then  have 

(y  +  3y-ll(y  +  3)-i-26^f-5y  +  2_l      5   ^   2 

f  f  y    v'    f 

Eeplaciug  ?/  by  cc  —  3,  the  result  takes  the  form 

_J^ 5  2 

x-3      {x-3y      {x-3y 

This  shoAVS  that  the  given  fraction  can  be  expressed  as 
the  sum  of  three  partial  fractions,  whose  numerators  are 
independent  of  x,  and  whose  denominators  are  the  powers 
of  ic  —  3  beginning  with  the  first  and  ending  with  the  third. 

Similar  considerations  hold  with  respect  to  any  example 
under  Case  II. ;  the  number  of  partial  fractions  in  any  case 
being  the  same  as  the  number  of  equal  factors  in  the 
denominator  of  the  given  fraction. 

EXAMPLES. 

(5  a;  -4-  5 

476.  1.    Separate  — into  partial  fractions. 

(3a; +  5)- 


UNDETERMINED  COEFFICIENTS.  337 

In  accordance  with  the  principle  stated  in  Art.  475,  we 
assume  the  given  fraction  equal  to  the  sum  of  tioo  partial 
fractions,  whose  denominators  are  the  powers  of  3  a;  +  5  be- 
ginning with  the  first  and  ending  with  the  second;  that  is, 

6.T  +  5    _     A  B 


(3x  +  5)2     3a; +  5      (3a; +  5)' 
Clearing  of  fractions,  we  have 

Qx  +  5  =  A{:5x  +  b)  +  B 

Equating  the  coefficients  of  like  powers  of  x, 
3^  =  6, 
and  bA+B  =  b. 

Solving  these  equations,  we  have  A  =  2  and  J5  =  —  5. 

^^''''''^'       .  (3a; +  5)^ "  3^  ~  (3a; +  5)2' 
Separate  each  of  the  following  into  partial  fractions : 

2  2a; -13  ^      3a;^-4  g     a;(5a;-4) 

•    a;-+10a;  +  25'  '    {x  +  lf  '     (5a;- 2)^' 

x^  5     18.a;^+12a;-3  ^     x{x  +  2y 

'    {x-2y  '         (3a; +  2)^     '  '     (x  +  iy' 

„     2ar''-10a;2  +  17a;-10  g     4.x'- IRx" 

{x-iy         '        '   {2x-3y' 

Case  III. 

477.    When   some  of  the  factors  of  the  denominator  are 
equal. 

1.    Separate  — ^  into  partial  fractions. 

^  a;(a;  +  l)'^ 


338  COLLEGE  ALGEBRA. 

The  method  in  Case  III.  is  a  combination  of  the  methods 
of  Cases  I.  and  II. ;  we  assume 

3.  +  2    _A^B^^C_I>  (1) 


x{x  +  iy    X    x  +  i    {x  +  iy    {x  +  i) 

Clearing  of  fractions, 

3x  +  2  =  A{x  +  lY-{-Bx{x  +  ly  +  Cx{x  +  1)  +  Dx 

=  {A  +  B)x'  +  {^A  +  2B+C)x' 

+  {'5A  +  B+C  +  D)x  +  A. 

Equating  the  coefficients  of  like  powers  of  x, 

A  +  B  =  0, 

3^  +  25  +  C=0, 

^A  +  B  +  G  +  D  =  d, 

and  A  =  2. 

Solving  these  equations,  we  have 

yl  =  2,  5=  -  2,  C=  -  2,  and  i)  =  1. 

Substituting  in  (1), 

_3^_-f2_  ^  2  _  _2 2  1 

x{x  +  iy'     X     re  4-1      {x  +  iy     {x  +  iy 

Note.  It  is  impracticable  to  give  an  illustrative  example  for  eveiy 
pdSsible  case ;  but  the  student  will  find  no  difficulty  in  assuming  the 
proper  partial  fractions  if  attention  is  given  to  the  following  general 
rule : 

A  fraction  of  the  form should  be  put 

equalto  (x  +  a)(x  +  6)...(x  +  m)-.. 

x+a     x  +  b  x  +  m      {x  +  my  {x  +  m)'^ 

Single  factors  like  x  +  a  and  x  +  h  having  single  partial  fractions 
corresponding,  arranged  as  in  Case  I.  ;  and  repeated  factors  like 
(x  +  my  having  r  partial  fractions  corresponding,  arranged  as  in 
Case  II. 


UNDETERMINED  COEFFICIENTS.  339 

EXAMPLES. 
Separate  each  of  the  following  into  partial  fractions : 

x{x  +  2y'  '       x{x-l){x-'2y 

3       3.T-1  g     IB-Tx  +  Sx'-Sx' 

ar(a;  +  l)-  *  x*-\-5a^ 

^  3a^  —  7x  +  3  -    ,^     5x^-{-3x-{-2 


(2a;-3)(2a^  — Tx  +  O)  x^{x-\-iy 

478.  If  the  degree  of  the  numerator  is  equal  to,  or  greater 
than,  that  of  the  denominator,  the  preceding  methods  are 
inapplicable. 

x'^  —  3  x^  —  1 
Let  it  be  required,  for  example,  to  separate ^ 

into  partial  fractions.  '*'  ~ 

If  we  proceed  as  in  Case  I.,  we  should  assnme 


0?  —  X  X        x  —  1 

Clearing  of  fractions  and  uniting  terms, 

or^  -  3ar  -  1  =  (^1  +  B)x  -  A. 
Equating  the  coefficients  of  x^,  we  have  1  =  0;  a  result 
which  shows  that  the  method  of  Case  I.  is  inapplicable. 
But  by  actual  division  we  obtain 

ar  —  X  X-  —  X 

We  can  now  separate  — '^ into  partial  fractions  by 

XT  —  X 

the  method  of  Case  I.  ;  the  result  is 

1 3_ 

X      x—i 
Substituting  in  (1),  we  have 

X-  —  X  X       X  —  1 


340  COLLEGE   ALGEBRA. 


EXAMPLES. 


Separate  each  of  the  following  into  an  integral  expression 
and  two  or  more  partial  fractions  : 

J        8x^-36x^-2  g     5x^  +  5x^-2x^  +  3 

(2.r-5)(2a;  +  l)"  '  x*  +  cc' 

'     2     3ar^  +  19a;^  +  35a;  '    ^     3af -2x' +  22x^ +  9x 

{x  +  2y         '  •  (ar-ir 


479.  If  the  denominator  of  a  fraction  can  be  resolved' 
into  factors  partly  of  the  first  and  partly  of  the  second,  or 
all  of  the  second  degree,  in  x,  and  the  numerator  is  of  a 
lower  degree  than  the  denominator,  the  Theorem  of  Unde- 
termined Coefficients  enables  us  to  express  the  given  frac- 
tion as  the  sum  of  two  or  more  partial  fractions,  whose 
denominators  are  factors  of  the  given  denominator,  and 
whose  numerators  are  independent  of  x  in  the  case  of  frac- 
tions corresponding  to  factors  of  the  first  degree,  and  of  the 
form  Ax  +  B  in  the  case  of  fractions  corresponding  to  fac- 
tors of  the  second  degree. 

The  only  exceptions  occur  when  the  factors  of  the  denom- 
inator are  of  the  second  degree  and  all  equal. 

1.    Separate  — into  partial  fractions. 

The  factors  of  the  denominator  are  x  +  1  and  xF  —  x  +  1. 

Assume  then,  —^  =  -^—  +    ^^+G  .  (V) 

'  x'  +  l      x  +  l^x'-x  +  l  ^  ' 

Clearing  of  fractions,  we  have 

1  =  A{x'  -x  +  l)  +  (Bx  +  C)  (:r  +  1) 

=  (.1  +  B) .r  +  ( -  yl  +  B  +C)x  +  A  +  C. 


UNDETERMINED   COEFFICIENTS.  341 

Equating  the  coefficients  of  like  powers  of  x, 
A  +  B  =  0, 
-A  +  B-\-C=0, 
and  A+C=l. 

Solving  these  equations,  A  =  -,  B  =  —  ~,  and  C=  -• 
o  O  3 

Substituting  in  (1),  Ave  have 

1  1  x-2 


a;3  +  l      3(a;-f-l)      ^{x'-x-\-l) 

EXAMPLES. 

Separate  each  of  the  folloAving  into  partial  fractious : 

x^-1  x^+x'  +  l 

^  ar^4-2a;-2  "     g     45  +  36a;-a;^ 

(x2  +  2)(a;2  +  a;  +  2)'  '    a-"  -  6ar'- 27* 

4_    20a;^-2a;^  ^     a;^  _  2.t^  + .r^- a;  +  1 

REVERSION  OF   SERIES. 

Note.   To  revert  a  given  series  y  =  a  +  bx"'  +  ex"  +  •••  is  to  express 
X  in  tlie  form  of  a  series  proceeding  in  ascending  powers  of  y. 

480.   Example.  Revert  the  series 

y  =  2x  +  x^-2a?-3x'-\ 

Assume  x  =  Ay+  By-  +  (7/  +  Dy"^  -\ .  (1  )- 

Substituting  in  this  the  given  value  of  y,  we  have 

x  =  A{2x  +  x- -23? -Sx' +'...) 

+  B{4x^  +  x'  +  437=  -  8a;^  +  ...) 

+  C(8a.-3  +  12a;^+ ...) 

-\-D{mx' +...)+.... 


342 


COLLEGE   ALGEBRA. 


That  is,  a;  =  2yla;+    A 
+  45 


x'-2A 

x'-   SA 

+  AB 

-    7B 

+  80 

+  12C 

+  16D 

x^  +  .... 


Equating  the  coefficients  of  like  powers  of  x. 
2A=1 
A+4:B=0 
-2A  +  4.B-hSC=0 
-3A-7B  +  12C+16D  =  0;  etc. 
Solving  these  equations, 

A  =  -,  B=--,  (7  =  -^,  i)  =  -il,  etc. 

2  8  16  128 


Substituting  in  (1),  we  have 

1   ,  ,    3    .       13 


2^      8^       16^ 


128 


y*  + 


If  the  even  powers  of  x  are  wanting  in  the  given  series, 
the  operation  may  be  abridged  by  assuming  x  equal  to  a 
series  containing  only  the  odd  powers  of  y. 

Thus,  to  revert  the  series  y  =  x  — x^  +  x"  — x^-^---,  we 
should  assume 

x  =  Ay  +  Bf  +  Cf  +  Df  +  ■'•  • 

If  the  odd  powers  of  x  are  wanting  in  the  given  series, 
the  reversion  of  the  series  cannot  be  effected  by  the  method 
previously  given.  But  by  substituting  another  letter,  say  t, 
for  X-,  we  may  revert  the  series  and  express  t  in  terms  of  y ; 
and  by  taking  the  square  root  of  the  result,  x  itself  may  be 
expressed  in  terms  of  y. 

If  the  first  term  of  the  given  series  is  independent  of  x, 
it  is  impossible,  by  the  method  previously  given,  to  express 
X  in  the  form  of  a  series  proceeding  in  ascending  powers  of 
y;  but  it  is  ])()ssible  to  ex])ress  it  in  the  form  of  a  series  in 
Avhich  y  is  the  uiily  uukuowu  quantity. 


UNDETERMINED   COEFFICIENTS.  343 

Let  it  be  required,  for  example,  to  revert  the  series 

y  =  2-\-2x  +  x^-2x^-3x^+---  • 
Tlie  series  may  be  written 

y-2  =  2x  +  x'-2x'-SxU 

We  then  assume 

X  =  A(y  -  2)  +B(y  -  2y+C(y  -  2y+D{y  -  2)^+  ••.  • 
Proceeding  as  before,  we  find 

x  =  ^{y-2)-^(y-2Y+  —  (y-2Y-—(y-2Y  +  '-'' 

EXAMPLES. 
481.   Revert  each  of  the  following  to  four  terms  : 
f  l/l    y  =  X  -{■  '£-  -\-  X?  -\-  x^  -\-  '• '  • 
^     2.    ?/  =  3x--2x2+3af'-4a;''+.... 

^2       4         6         8 

^         i  i_3  li 

-      5.    y  =  x  —  7?-\-x'  —  '3?-\--"' 

6.  y  =■ u  ...  . 

^      2      3       4       5^ 

7.  2/  =  3a;  +  5a^  +  7af'  +  lla;^  +  o.oo 

/y»3  /y*5  /y,7 


344  COLLEGE   ALGEBRA. 

XXXIII.    THE  BINOMIAL  THEOREM. 

FRACTIONAL   AND   NEGATIVE   EXPONENTS. 

,482.    It  was  proved  in  Art.  445  that,  when  n  is  a  positive 

integer, 

/^t    .     \n      1    ,  ,   n(n  —  l)    o  ,  n(n  —  l)(n  —  2)    n   , 

(1  +  0;)"=  1  +nx  +  -^-— — ^-x^  +  ^ f;^ ^ar^  +  .... 

PROOF  OF  THE  THEOREM  FOR  FRACTIONAL  OR 

NEGATIVE  EXPONENTS. 

Note.  We  shall  use  the  expression  "Fractional  or  Negative  Ex- 
ponent," in  the  present  chapter,  to  signify  a  rational  exponent  which 
is  not  a  positive  integer. 

483.    I.  Wheyi  the  exponent  is  a  positive  fraction. 

Let  the  exponent  be  -,  where  p  and  q  are  positive  integers. 


By  Art.  282,   (1  +  x)^  =  ^{1  +  xy 


=  ^l+px  +  --  (Art.  482). 

In  Art.  263,  we  gave  a  rule  for  extracting  the  ?ith  root  of 
a  polynomial  which  is  a  perfect  power  of  the  nth  degree. 

It  is  evident,  therefore,  as  in  Art.  449,  that  -y/l+jix  -\ 

can  be  expanded  in  a  series  proceeding  in  ascending  powers 
of  X ;  thus, 

px 
l+pa;  +  ...  1+'— +  ••• 


q   \  2^x-\-  ••• 

p  vx 

That  is,  (i  +  a;)»  =  l+Y  +  -...  (1) 


THE   BINOMIAL   THEOREM.  345 

II.  When  the  exponent  is  a  negative  integer  or  a  negative 
fraction. 

Let  the  exponent  be  —  s,  where  s  is  a  positive  integer  or 
positive  fraction. 

By  Art.  284,  (l  +  x)-' 


{i  +  xy 
1 

1  +  sx-Jr 


-,  by  Art.  482  or  Case  I. 


It  is  evident,  as  in  Art.  449,  that  can  be  ex- 

'  1+SX+--- 

paneled,  by  actual  division  in  a  series  proceeding  in  ascending 
powers  of  x ;  thus, 

l  +  sx  +  ---)l{l—sx-\ 

l  +  sx-\ 

—  sx—  ••• 
That  is,     '         (1  +  x)-' =  1  -  saj  H (2) 

From  (1)  and  (2)  we  observe  that,  when  n  is  fractional 
or  negative,  the  form  of  the  expansion  is 

(1  4-  xy  ^l  +  nx  +  A^-  +  Bx^  ^ (3) 

Writing  -  in  place  of  x,  we  obtain 
a 

aj  a,  a-         a'^ 

Multiplying  both  members  by  a", 

(a  +  a-) "  =  a"  +  na"'^ x  +  Aa''~^ ar  +  Ba"'^ x^  -\ •   (4) 

This  result  is  in  accordance  with  the  second,  third,  and 
fourth  laws  of  Art.  443 ;  hence  these  three  laws  hold  for 
fractional  or  negative  values  of  the  exponent. 

We  will  now  prove  that  the  ffth  law  of  Art.  443  holds  for 
fractional  or  negative  values  of  the  exponent. 

Let  P  and  Q  denote  the  coefficients  of  x''  and  a;''+^,  respec- 
tively, in  the  second  member  of  (3). 


346  COLLEGE   ALGEBRA. 

Then  (3)  and  (4)  may  be  written 

(l-\-xy  =  l+nx-] \-Px''-{-  Qx''+''  +  ...,  f5) 

and     {a  +  x)"  =  a" -i-na"^x-^  ••• 

+  Pa"-'-a;'-  +  Qa"-'-'a;'-+^H (6) 

In  (6)  put  a  =  l  +  y  and  x  =  z;  then, 

(l+2/  +  2:)»=(l  +  2/)"+...+P(l  +  2/)"-'-2'-  +  ....        (7) 

Again,  in  (5)  put  x  =  z  +  y;  thus, 

(l  +  z-^yr=l  +  -+P{z+yy+Q{z  +  7jy+'  +  .^.. 

Expanding  the  powers  of  z  +  yhy  aid  of  (6),  we  have 

(1  +  z  +  2/)"  =  1  +  ...  +  P[z^  +  rz^-'y  +  .-] 

+  Q[,^+i+(r+l),^y  +  . ..]+....  (8) 

The  first  members  of  (7)  and  (8)  being  identical,  their 
second  members  must  be  equal  for  every  value  of  z  which 
makes  both  series  convergent  (Art.  452)  ;  and  by  the  Theorem 
of  Undetermined  Coefficients,  the  coefficients  of  z''  in  the 
two  series  are  equal ;  that  is, 

P(l +  ?/)"-' =  P+Q(r +1)2/  + terms  in  y%  f,  etc. 

Expanding  the  first  member  by  aid  of  (5),  this  becomes 

P[l+(,i_r)r/  +  ...]  =  P+Q(r  +  l)i/+.... 

This  equation  is  satisfied  by  every  value  of  y  which  makes 
both  members  convergent,  and  hence  the  coefficients  of  y  in 
the  two  series  are  equal ;  that  is, 

P(n  -  r)  =Q{r  +  1),  or  Q  =  P>^zJl. 
r+  1 

But  in  the  second  member  of  (G),  n  —  r  is  the  exponent 
of  a  in  the  texm  whose  coefficient  is  P,  and  r  +  1  is  the  ex- 
ponent of  X  in  that  term  increased  by  1. 

Therefore  the  fifth  law  of  Art.  443  is  proved  to  hold  for 
fractional  or  negative  values  of  the  exponent. 


THE   BINOMIAL   THEOREM.  347 

By  aid  of  the  fifth  law,  the  coefficients  of  the  successive 
terms  after  the  second,  in  the  second  member  of  (6),  may 
be  readily  found  as  in  Art.  444;  thus, 

(a  +  xy  =  a"  +  na^-'x  +  "^^^—^a"-':^ 

,  n(n  —  l)(n  —  2)    _  ,   ,,  ^f^\ 

The  second  member  of  (9)  is  an  infinite  series ;  for  if  n 
is  fractional  or  negative,  no  one  of  the  quantities  n  —  1, 
n  —  2,  etc.,  can  become  equal  to  zero. 

The  result  expresses  the  value  of  (a  +  a;)"  only  for  such 
values  of  a  and  x  as  make  the  series  convergent  (Art.  452). 

Note.   Divicung  both  members  of  (9)  by  a",  we  have 

(l  I  ^V=l  1  n^  1  »(»-l)^^  I  n(n-l)(n-2)x^  ^ 
\       aj  a  \2        a'^  [3  a^ 


numerically  less  than  1 ;  hence  the  series  (9)  is  convergent  when  x  is 
numerically  less  than  a. 

EXAMPLES. 

484.  In  expanding  expressions  by  the  Binomial  Theorem 
when  the  exponent  is  fractional  or  negative,  the  exponents 
and  coefiicients  of  the  terms  may  be  obtained  by  aid  of  the 
laws  of  'Art.  443,  which  have  been  proved  to  hold  for  all 
rational  values  of  the  exponent. 

Notes  3  and  4,  Art.  446,  apply  with  equal  force  to  the 
examples  in  the  present  article. 

1.    Expand  (a  +  a;)^  to  four  terms. 

2 
The  exponent  of  a  in  the  first  term  is  -,  and  decreases  by 

1  in  each  succeeding  term. 


348  COLLEGE   ALGEBRA. 

The  exponent  of  x  in  the  second  term  is  1,  and  increases 
by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1 ;  of  the  second  term, 

2  2  1 
- ;  multiplying  -  the  coefficient  of  the  second  term,  by  —  -, 

3  3  o 

the  exponent  of  a  in  that  term,  and  dividing  the  product  by 
the  exponent  of  x  increased  by  1,  or  2,  we  have  —  t:  as  the 
coefficient  of  the  third  term ;  and  so  on.     Hence, 

(^a  +  x)'^  =  J  +  la-ix-la-^oc^  +  ^a-'^x' 

o  y  oi 


2.    Expand  (1  —  2  a;  ^)-no  five  terms. 
(l-2i»-^)-2  =  [l+(-2x-^)]-2 

=  l-2_  2.1-3. (_2a;-*)  +  3.1-^.(-2a;-')2 

-4:.l-''{-2x-^y+5-l-'-{-2x-fy- 

=  l-\-4:X~^  +  12x-^  +  32x--^  -\-80x~^-i 


3.    Expand    ^  —  to  five  terms. 


Va-i  +  Sx*      {a-'  +  3xs)^ 


[(«-)  + (3x0] 


hi-i 


-f^ia-r'^'isxh^ 

=ai-Jx^+2a^x^-^a^x+^a'^'x^+... 
o  o 


THE   BINOMIAL   THEOREM.  349 

Expand  each,  of  the  following  to  five  terms  : 
V  4.    (a  +  x)  I  -    12.    {a'-2x-')- 1 


13. 


5. 

il  +  x)-\ 

i  6. 

(i-x)-i 

7. 

</a-x. 

-/   8. 

1 

</l  +  x 

9. 

1 

{a-xf 

^10. 

{x^-^y)\ 

11. 

(«-*HT' 

ar  +  4y 
14.    (.T-3  +  2a&)l 
1 


15 


(a-i-3r*)= 


16.  (l+^)-^ 


17.    V(4a-  +  x-3)5. 

18.  ^J^_2^;r^V'. 


485.  The  formula  for  the  rth  term  of  (a  +  a;)»  (Art.  447) 
holds  for  fractional  or  negative  values  of  n,  since  it  was 
derived  from  an  expansion  which  has  been  proved  to  hold 
for  all  rational  values  of  the  exponent. 

EXAMPLES. 
1.   Findthe  seventh  term  of  (a  —  3x~^)~^ 
(a-3a;"^)"^  =  [a  +  (-3a;"^)]"i 

In  this  case,  r  =  7  and  n  =  —  - ;  hence  the  seventh  term 
3 


a-¥(_3a;-f)6 


1 

3' 

4        7        10        13 
3 '      3  '       3  "       3  ' 

16 
3 

1. 2.3. 4. 5-6 

=  fa-V(3=.-.)  =  ipcr-V.- 


350  COLLEGE   ALGEBRA. 

Note.    Notes  3  and  4,  Art.  446,  apply  with  equal  force  to  the 
examples  in  the  present  article. 

Pind  the 
•j,  2.    Eighth  term  of  (a  +  x)^. 

3.    Twelfth  term  of  (1  +  m)-\ 

-'  4.    Fifth  term  of  (1  -a^)"! 

9 

5.  Sixth  term  of  (a  —  xy. 

i  6.  Seventh  term  of  {a^-\-b^)~^. 

7.  Eighth  term  of  {x-^-y~^)^. 

-4  8.  Sixth  term  of    . 

9.    Eleventh  term  of  (a" ^  +  2a;)*. 
1 


10.    Ninth  term  of 


(71-^ -ct)' 


._i 


11.  Sixth  term  of  (a'+ox-^)' 

12.  Tenth  term  of  (xVy  —  -^J    • 

13.  Find  the  term  involving  x"^  in  f  af'  -  - 

T:    14.  Find  the  term  involving  cc »  infa;  ^4-— -3)     •     j 

486.    To  find  any  root  of  a  number  ax>proximaiely  by  the 
Binomial  Theorem. 

1.    Find  the  approximate  value  of  ^25  to  five  places  of 

decimals. 

■^25  =  25'^  =  (27  -  2)  3  =  (3^  -  2)  J. 


THE   BINOMIAL   THEOREM.  351 

Expanding  by  the  Binomial  Theorem,  we  have 
[(3'^)  +  (-2)]^  =  (3^)^  +  3(3TH-2)-^(3T^(-2)^ 
+  A(33)-I(_2r-... 
=  3.       ^  4  40 


3.32     9.3^     81-3« 

Expressing  the  value  of  each  fraction  approximately  to 
the  nearest  fifth  decimal  place,  we  have 

^/25  =  3  -  .07407  -  .00183  -  .00008 

=  2.92402. 


Separate  the  given  number  into  two  parts,  the  first  of  which 
is  the  nearest  2')erfect  power  of  the  same  degree  as  the  required 
root. 

Expand  the  result  by  the  Binomial  Theorem. 

Note.  If  the  second  term  of  the  binomial  is  small  compared  with 
the  first,  the  terms  of  the  expansion  diminish  rapidly ;  but  if  the 
second  term  is  large  compared  with  the  first,  it  requires  a  great  many 
terms  to  ensure  any  degree  of  accuracy. 

EXAMPLES. 

Find  the  approximate  value  of  each  of  the  following  to 
five  places  of  decimals  : 

2.  VlO.  4.    </7.  6.    ^yi5. 

3.  Viol.  5.    a/84.  7.    a/28. 


352  COLLEGE   ALGEBRA. 


XXXIV.    LOGARITHMS. 

487.  Every  positive  real  number  may  be  expressed, 
exactly  or  approximately,  as  a  power  of  10 ;  thus, 

100  =  10^;  13  =  10i"3«-;  etc. 

When  thus  expressed,  the  corresponding  exponent  is 
called  its  Logarithm  to  the  base  10;  thus,  2  is  the  logarithm 
of  100  to  the  base  10,  a  relation  which  is  written 

logio  100  =  2,  or  simply  log  100  =  2. 

And  in  general,  if  lO""  =  m,  then  x  =  log  m. 

488.  Logarithms  of  numbers  to  the  base  10  are  called  Co7n- 
mon  Logarithms,  and,  collectively,  form  the  Common  System. 

They  are  the  only  ones  used  for  numerical  computations. 

Any  positive  real  number,  except  unity,  may  be  taken  as 
the  base  of  a  system  of  logarithms  ;  thus,  if  a'  =  m,  where 
a  and  m  are  positive  real  numbers,  then  x  =  log„m. 

Note.    A  negative  mimber  is  not  considered  as  having  a  logarithm. 

489.  By  Arts.  283  and  284, 

100=1,  10-^: 

10^  =  10,  10--: 

102=100,  10-3: 

Whence,  by  the  definition  of  Art.  487, 

logl  =  0,  log  .1  =  -1  =  9-10, 

log  10  =  1,  log  .01  =  -  2  =  8  -  10, 

log  100  =  2,  log  .001  =  -  3  =  7  -  10,  etc. 

Note.  The  second  form  for  log.1,  log. 01,  etc.,  is  preferable  in 
liractiee.      Where  no  base  is  expressed,  the  bast-  10  is  imderstood. 


=.1, 

1 

.00 

=  .01, 

J_ 

/WW 

.  =  .001,  etc. 

LOGARITHMS.  353 

490.  It  is  evident  from  Art.  489  that  the  logarithm  of 
a  mimber  greater  than  1  is  positive,  and  the  logarithm  of  a 
number  between  0  and  1  negative. 

491.  If  a  number  is  not  an  exact  power  of  10,  its  com- 
mon logarithm  can  only  be  expressed  approximately ;  the 
integral  part  of  the  logarithm  is  called  the  characteristic, 
and  the  decimal  part  the  mantissa. 

For  example,  log  13  =  1.1139. 

In  this  case  the  characteristic  is  1,  and  the  mantissa  .1139. 

For  reasons  which  which  will  be  given  hereafter,  only 
the  mantissa  of  the  logarithm  is  given  in  a  table  of  loga- 
rithms of  numbers  ;  the  'characteristic  must  be  supplied  by 
the  reader.      (Compare  A.rts.  492  and  493.) 

492.  It  is  evident  from  the  first  column  of  Art.  489,  that 
the  logarithm  of  a  number  between 

1  and      10  is  equal  to  0  plus  a  decimal ; 
10  and    100  is  equal  to  1  plus  a  decimal ; 
100 'and  1000  is  equal  to  2  plus  a  decimal ;   etc. 

Therefore,  the  characteristic  of  the  logarithm  of  a  num- 
ber with  one  figure  to  the  left  of  the  decimal  point,  is  0 ; 
Avith  two  figures  to  the  left  of  the  decimal  point,  is  1 ;  with 
three  figures  to  the  left  of  the  decimal  point,  is  2 ;  etc. 

Hence,  the  characteristic  of  the  logarithm  of  a  number  greater 
than  1  is  1  less  than  the  number  of  places  to  the  left  of  the  deci- 
mal j^oint. 

For  example,  the  characteristic  of  log  906328.51  is  5. 

493.  In  like  manner,  the  logarithm  of  a  number  between 
1  and      .1  is  equal  to  9  plus  a  decimal  —  10  ; 

.1  and    .01  is  equal  to  8  plus  a  decimal  —  10 ; 
.01  and  .001  is  ucpial  to  7  plus  a  decimal  —  10  ;  etc. 


354  COLLEGE   ALGEBRA. 

Therefore,  the  characteristic  of  the  logarithm  of  a  deci- 
mal with  no  ciphers  between  its  decimal  point  and  first 
significant  figure,  is  9,  with  — 10  after  the  mantissa ;  of  a 
decimal  with  one  cipher  between  its  point  and  first  figure  is 
8,  with  —  10  after  the  mantissa ;  of  a  decimal  with  tivo 
ciphers  between  its  point  9,nd  first  figure  is  7,  with  — 10 
after  the  mantissa ;  etc. 

Hence,  to  find  the  characteristic  of  the  logarithm  of  a  number 
less  than  1,  subtract  the  riumber  of  cijyhers  between  the  decimal 
point  and  first  significant  figure  from  9,  writing  —10  after  the 
mantissa. 

For  example,  the  characteristic  of  log  .00702.3  is  7,  with 
—  10  written  after  the  mantissa. 

Note.  Some  writers  combine  the  two  portions  of  the  characteristic, 
and  write  the  result  as  a  negative  characteristic  before  the  mantissa. 

Thus,  instead  of  7.G036  -  10,  the  student  will  frequently  find  3.6036, 
a  minus  sign  being  written  over  the  characteristic  to  denote  that  it 
alone  is  negative,  the  mantissa  being  always  positive. 

PROPERTIES    OF   LOGARITHMS. 

494.  In  any  system,  the  logarithm  of  unity  is  zero. 

For  by  Art.  283,  a°  =  1 ;  whence,  by  Art.  488,  log„l  =  0. 

495.  In  any  system,  the  logarithm  of  the  base  itself  is  unity. 
For,  a^  =  a;  whence,  log„a  =  1. 

496.  In  any  system  whose  base  is  greater  than  unity,  the 
logarithm  of  zero  is  minus  infinity. 

For  if  a  is  >  1,  a""''  =  -4  =  -  =  0  (Art.  210). 

a        cc 

Whence,  by  Art.  488,  log„0  =  -  c». 

Note.  Fo  literal  meaning  can  be  attached  to  such  a  result  as 
logaO  =  — co;   it  must  be  interpreted  as  follows: 

If,  in  any  system  whose  base  is  greater  than  unity,  a  number 
ji]ii)ro.T,ches  tlic  limit  0,  its  logarithm  is  negative,  and  increases  without 
limit  ill  ab.S(jiutf  value.      (^Cmiiiiare  Note  to  Ait.  212.) 


LOGARITHMS.  355 

497.  In  any  system  whose  base  is  less  than  unity,  the  logo- 
rithm  of  zero  is  infinity. 

For  if  a  is  <  1,  a°°  =  0  ;  whence,  log^O  =  oo. 

498.  In  any  system,  the  logarithm  of  a  i^oduct  is  equal  to 
the  sum  of  the  logarithms  of  its  factors. 

Assume  the  equations 

"*'  =  ^  I ;  whence  by  Art.  488,  |  ^  =  ^°^«^' 
a^  =  n  )  (y  =  loga^i. 

Multiplying,    a""  x  a"  =  inn,  or  a'^+^  =  mn. 

Whence,  log^  ??i«  =  x-\-y 

=  log„m  +  log„7t. 

In  like  manner,  the  theorem  may  be  proved  for  the  product 
of  three  or  more  factors. 

499.  By  aid  of  the  theorem  of  Art.  498,  the  logarithm  of 
any  composite  number  may  be  found  when  the  logarithms 
of  its  factors  are  known. 

1.    Given  log2  =  .3010,  and  logS  =  .4771 ;  find  log  72. 
log72  =  log(2x  2x2x3x3) 

=  log2  +log2  +  log2  +log3  +  log3 
=  3xlog2  +  2xlog3 
=  .9030  +  .9542  =  1.8572. 


EXAMPLES. 

Given     log2  =  .3010,     log3  =  .4771,     log5  =  .G990,    and 
log 7  =  .8451;  find: 

2.  log42.  —  6.    logll2.       10.    logl47.  14.  log514.5. 

3.  log  45.        7.    log  144.       11.    log  375.  15.  log  6048. 

4.  log63.        8.    log216.       12.    logGSG.  16.  Iogl2005. 

5.  Iogl05.     9.    logl35.       13.    logll34.  17.  logl5876. 


356  COLLEGE    ALGEBRA. 

500.  In  any  system,  the  logarithm  of  a  fraction  is  equal  to 
the  logarithm  of  the  numerator  minus  the  logarithm  of  the 
denominator. 

Assume  the  equations 

«^=^n;  whence,  j-'^^J^-^' 
a^  =  n  )  yy  =  log„7i. 

Dividing,  we  have        —  =  — ,  or  a''^^  =     • 
a"      n  n 

Whence,  log"  —  =  x  —  v 

n 

=  logo  m  —  log„n. 

501.  1.  Given  log2  =  .3010  ;  find  log  5. 

Iog5  =  log^=logl0-log2 

=  1  -  .3010  =  .6990. 


EXAMPLES. 
Given  log2  =.3010,  h)g 3  =.4771,  and  log 7=. 8451 ;  find: 

2.  log^.  5.    log35.       ^.    logi^.  11.    log57f. 

3.  log^'.  --'     6.    log||.  9.    logCf.     ^12.    log^. 

4.  logllli.      7.    log225.~^     10.    log245.        13.    log21x. 

502.  In  any  system,  the  logarithm  of  any  x>oiver  of  a  quan- 
tity is  equal  to  the  logarithm  of  the  quantity  multiplied  by  the 
exponent  of  the  jiower. 

Assume  the  equation 

a^  =  m;  whence,  x  =  log„  m. 
Raising  both  members  to  the  pth  power, 

a''-'  =  m'' ;   whence,  log„  m''  =  px  —p  log,///*. 


LOGARITHMS.  357 

503.  In  any  system,  the  logarithm  of  any  root  of  a  quantity 
is  equal  to  the  logarithm  of  the  quantity  divided  by  the  index 
of  the  root. 

-  i        1 

For,  \og^-{/m  =  \og^(m'')  =  -\og^m  (Art.  502). 

504.  1.   Given  log  2  =  .3010;  find  log2l 

log23  =  f  X  log2  =  ^  X  .3010  =  .5017. 
o  o 

Note.  To  multiply  a  logarithm  by  a  fraction,  multiply  first  by  the 
numerator,  and  divide  the  result  by  the  denominator. 

2.    Given  log 3  =  .4771;  find  log  ^/^ 

log  ^3-=^  = --^=.0596.  . 


EXAMPLES. 
Given  log 2  =  .3010,  log 3  =.4771,  and  log 7=  .8451 ;  find. 

3.  logSl    "-e.   logl4^        9.   log253-.        12.   log  a/5. 

4.  log7«.       7.   log  12.1.;' 10.    logV7.        13.   logv^. 

5.  log5i      8..  logl5i      11.   log  a/2.        14.    log  a/126. 


15.    Find  log  (23  X  30- 

By  Art.  498,  log  (2^  x  3^)  =  log 2'  +  log3T 


^log2  +  -log3=.6967. 


"3     °      '  4     ^^^ 

Find  the  logarithms  of  the  following : 

^(fj-  "^«>S-  %4 

...  ^f . 

17.    3^^x21          19.    3</7.          21.   ^• 

23.    ^«. 

105 

358  COLLEGE   ALGEBRA. 

505,  In  the  common  system,  the  viantissce  of  the  logarithms 
of  numbers  having  the  same  sequence  of  figures  are  equal. 

Suppose,  for  example,  that  log  3.053  =  .4847  ;  then, 

log  305.3  =  log  (100  X  3.053)  =  log  100  +  log  3.053 

=  2 +  .4847  =2.4847; 

log  .03053  =  log  (.01  X  3.053)  =  log  .01  +  log  3.053 

=  8  -  10  +  .4847  =  8.4847  -  10 ;  etc. 

In  general,  if  n  is  any  positive  or  negative  integer, 

log  (10"  X  ??i)  =  7iloglO  +  log??i  =  n  +  logm. 

But  10"  X  m  is  a  number  which  differs  from  m  only  in 
the  position  of  its  decimal  point,  and  n  +  log  m  is  a  num- 
ber having  the  same  decimal  part  as  log7>i. 

Hence,  if  two  numbers  have  the  same  sequence  of  figures, 
the  mantissse  of  their  logarithms  are  equal. 

The  reason  will  now  be  seen  for  the  statement  made  in 
Art.  491,  that  only  the  mantissas  are  given  in  a  table  of 
logarithms.  For,  to  find  the  logarithm  of  any  number,  Ave 
have  only  to  take  from  the  table  the  mantissa  corresponding 
to  its  sequence  of  figures,  and  the  characteristic  may  then  be 
prefixed  in  accordance  with  the  rules  of  Arts.  492  and  493. 

Thus,  if  log  3.053  =  .4847,  then 

log  30. 53  =  1.4847,         log  .3053      =  9.4847  -  10, 

logS05.3  =  2.4847,         log  .03053    =8.4847-10, 

log 3053.  =  3.4847,         log  .003053  =  7.4847  -  10,  etc.  ^ 

This  property  is  only  enjoyed  by  the  common  system  of 
logarithms,  and  constitutes  its  superiority  over  others  for 
the  purposes  of  numerical  computation. 

506.  1.  Given  log 2  =  .3010,  log 3  =  .4771 ;  find  log. 00432. 
log 432  =  lug(2'  X  3')  =  1  log 2  +  3  log 3  ^  2.0353. 


LOGARITHMS.  35? 

Then  by  Art.  505,  the  mantissa  of  the  result  is  .6353. 
Whence  by  Art.  493,  log  .00432  =  7.6353  -  10. 

EXAMPLES. 
Given  log2  =  .3010,  log3  =  .4771,  and  log7  =  .8451 ;  find : 

2.  log  19.6.         ^^     6.   log 7350.  10.   log. 06174. 

3.  log4.8.  7.   Iog4.05.  11.   log(8.1)^ 

4.  log. 384.  8.    log. 000448.    ■        12.    log ^9600. 

5.  log.00315.  9.    log 302.4.  13.   log (22.4) «. 

507.    To  prove  the  relation 

T  log.m 

logjm  =  — ^"— 
log,.  0 

Assume  the  equations 

cC  =  m)        T  (a;  =  log„m, 

- ;  whence,  <  t^a    ) 

b''  =  m}  (i/  =  log^m. 

From  the  assumed  equations,  a"^  =  b^,  or  a"  =  b. 

Therefore,  log„&  =  -,    or  y  = 

y  '       log,,  6 

That  IS,  logjm 


By  aid  of  this  relation,  if  the  logarithm  of  a  quantity  m  to 
a  certain  base  a  is  known,  its  logarithm  to  any  other  base  b 
may  be  found  by  dividing  by  the  logarithm  of  b  to  the  base  a. 

A-  508.    To  prove  the  relation 

log^a  X  log„6  =  l. 
Putting  m  =  a  in  the  result  of  Art.  507,  we  have 

log.a  =  |^  =  -J-(Art.495). 
log,&      log„& 

Whence,  log^a  x  log„^  =  1. 


5G0 


COLLEGE   ALGEBRA. 


So. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

00S6 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

PS3I 

0569 

0607 

0645 

06S2 

0719 

0755 

12 

0792 

0828 

0S64 

0899 

0934 

0969 

1004 

1038 

1072 

1 106 

13 

"39 

"73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

264S 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

287S 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3S74 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

41 16 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

42S1 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5^32 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

55*52 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5<358 

5670 

37 

56S2 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

57S6 

38 

5798 

5S09 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6S93 

49 

6902 

691 1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7.8s 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

73S0 

7388 

7396 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITPIMS. 


361 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

55 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

75S9 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7S03 

7S10 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

79S0 

7987 

63 

7993 

8007 

8014 

8028 

8035 

8041 

8048 

TM 

64 

8062 

S069 

8075 

8082 

80S9 

8096 

8102 

8109 

8116 

65 

S129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

81S2 

81S9 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

S254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

S325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

838S 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

S579 

85S5 

8591 

8597 

8603 

8609 

8615 

8621 

l^/ol 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8S0S 

8S14 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8S71 

8876 

8S82 

8S87 

8S93 

8S99 

8904 

8910 

8915 

78 

8921 

8927 

8932 

S938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9UJ9 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

8i 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

91S6 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

937^ 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

94CO 

9425 

9430 

9435 

9440 

83 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9^57 

9661 

9666 

9671 

9675 

9680 

93 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

97-2 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9S23 

9S27 

9S32 

9836 

9841 

9845 

9S50 

9854 

9859 

9S63 

C7 

9S68 

9872 

9S77 

9S81 

9S86 

9890 

9894 

9899 

9903 

9908 

98 

9912  9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956  9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 
8 

9996 

2I0. 

0    1    1 

2 

3 

4 

5 

6 

7 

9 

362  COLLEGE  ALGEBRA. 

USE  OF  THE  TABLE. 

509.  The  table,  (i)ages  300  and  361)  gives  the  mantissa? 
of  tlie  logarithms  of  all  integers  from  100  to  1000,  calcu- 
lated to  four  places  of  decimals. 

510.  To  find  the  logarithm  of  a  number  of  three  figures. 
Look  in  the  column  headed  "No."  for  the  first  tAvo  sig- 
nificant figures  of  the  given  number. 

Then  the  mantissa  required  will  be  found  in  the  corre- 
sponding horizontal  line,  in  the  vertical  column  headed  by 
the  third  figure  of  the  number. 

Finally,  prefix  the  characteristic  in  accordance  with  the 
rules  of  Arts.  492  and  493. 

For  example,        log  1G8  =  2.2253  ; 

log  .344  =  9.5366 -10;  etc. 

511.  For  a  number  consisting  of  one  or  two  significant 
figures,  the  column  headed  0  may  be  used. 

Thus,  let  it  be  required  to  find  log  83  and  log  9. 
By  Art.  505,  log  83  has  the  same  mantissa  as  log  830,  and 
log  9  the  same  mantissa  as  log  900. 

Hence,       log  83  =  1.9191,  and  log  9  =  0.9542. 

512.  To  find  the  logarithm  of  a  number  of  more  thlin  three 
figures. 

1.    Required  the  logarithm  of  327.6. 

We  find  from  the  table,  log  327  =  2.5145, 
log  328  =  2.5159. 

That  is,  an  increase  of  one  unit  in  the  number  produces 
an  increase  of  .0014  in  the  logarithm. 

Therefore  an  increase  of  .6  of  a  unit  in  the  number  will 
produce  an  increase  of  .6  x  .0014  in  the  logarithm,  or  .0008 
to  the  nearest  fourth  decimal  place. 

Hence,      log  327.6  =  2.5145  -f  .0008  =  2.5153. 


LOGARITHMS.  363 

Note.  The  difference  between  any  mantissa  in  tlie  table  and  tlie 
mantissa  of  tlie  next  higher  number  of  three  figures,  is  called  the  tabu- 
lar difference.     The  subtraction  may  be  performed  mentally. 

The  following  rule  is  derived  from  the  above  : 

Fmd  from  the  table  the  mantissa  of  the  first  three  significant 
figures,  and  the  tabular  difference. 

Multiply  the  latter  by  the  remaining  figures  of  the  number, 
with  a  decimal  point  before  them. 

Add  the  residt  to  the  mantissa  of  the  first  three  figures,  and 
prefix  the  p>roper  characteristic. 

EXAMPLES. 

2.    Find  the  logarithm  of  .021508. 

Tabular  difference  =     21  Mantissa  of  215  =  3324 

■08  2 

Correction  =  1.G8  =  2,  nearly.  3326 

Eesult,  8.3326  - 10. 

Find  the  logarithms  of  the  following : 

3.  80.  7.    .7723.  11.   20.08.  15.  5.1809.    0 

4.  6.3.  8.    1056.  12.    92461.  >^  16.  1036.5.     ' 

5.  298.  9.    3.294.  13.    .40322.  -  17.  .086676. 

6.  ,902.        10.    .05205.  14.    .007178.  18.  .11507. 

513.   To  find  the  number  corresijonding  to  a  logarithm. 

1.    Required  the  number  whose  logarithm  is  1.6571. 

Find  in  the  table  the  mantissa  6571. 

In  the  corresponding  liiie,  in  the  column  headed  ''No./' 
we  find  45,  the  first  two  figures  of  the  required  number,  and 
at  the  head  of  the  column  we  find  4,  the  third  figure. 

Since  the  characteristic  is  1,  there  must  be  two  places  to 
the  left  of  the  decimal  point  (Art.  492). 

Hence,  the  number  corresponding  to  1.6571  is  45.4. 


364  COLLEGE   ALGEDRA. 

2.  Eeqiiired  the  number  whose  logarithm  is  2.3934. 

We  find  in  the  table  the  mantissse  3927  and  3945,  whose 
corresponding  numbers  are  247  and  248,  respectively. 

That  is,  an  increase  of  18  in  the  mantissa  produces  an 
increase  of  one  unit  in  the  number  corresponding. 

Therefore,  an  increase  of  7  in  the  mantissa  will  produce 
an  increase  of  -^^  of  a  unit  in  the  number,  or  .39,  nearly. 

Hence,  the  number  corresponding  is  247  +  .39,  or  247.39. 

The  following  rule  is  derived  from  the  above  : 

Find  from  the  table  the  next  less  mantissa,  the  three  figures 
corresponding,  and  the  tabular  difference. 

Subtract  the  next  less  from  the  given  mantissa,  and  divide 
the  remainder  by  the  tabular  difference. 

Annex  the  quotient  to  the  first  three  figures  of  the  number, 
and  x>oint  off  the  residt. 

Note.  The  rules  for  pointing  off  are  the  reverse  of  those  of  Arts. 
492  and  493 : 

I.  If  —  10  is  not  written  after  the  mantissa,  add  1  to  the  character- 
istic, giving  the  number  of  places  to  the  left  of  the  decimal  point. 

II.  If  — 10  is  written  after  the  mantissa,  subtract  the  positive  part  of 
the  characteristic  from  9,  giving  the  number  of  ciphers  between  the 
decimal  point  and  first  significant  figure. 

EXAMPLES. 

3.  Find  the  number  whose  logarithm  is  8.52G4  —  10. 

5264 

Next  less  mantissa,  5263 ;  three  figures  corresponding,  336. 
Tabular  difference,  13)  1.000 (.077  =  .08,  nearly. 
91 
90 

According  to  the  above  rule,  there  will  be  one  cipher 
between  the  decimal  point  and  first  significant  figure. 
Hence,  number  corresponding  =  .033608. 


LOGARITHMS.  365 

Find  the  numbers  corresponding  to  tlie  following  loga- 
rithms : 


-r  4. 

1.8055. 

9.    8.1648-10. 

14. 

1.6482. 

5. 

9.4487  - 

-10. 

+  10.    7.5209-10. 

■h  15. 

7.0450  - 

-10. 

-  6. 

0.21G5. 

11.   4.0095. 

16. 

4.8016. 

7. 

3.9487. 

12.   0.9774. 

■^7. 

8.1144  - 

-10. 

^8. 

2.7364. 

-f-lS.    9.3178-10. 
APPLICATIONS. 

18. 

2.7015. 

514.  The  approximate  value  of  an  arithmetical  quantity, 
in  which  the  operations  indicated  involve  only  multiplica- 
tion, division,  involution,  or  evolution,  may  he  conveniently 
found  by  logarithms. 

The  utility  of  the  process  consists  in  the  fact  that  addition 
takes  the  place  of  multiplication,  subtraction  of  division, 
multiplication  of  involution,  and  division  of  evolution. 

Note.  In  computations  with  four- place  logarithms,  the  results  can- 
not usually  be  depended  upon  to  more  than  four  significant  figures. 

515.  1.    rind  the  value  of  .0631  x  7.208  x  .51272. 
By  Art.  498,  log  (.0631  x  7.208  x  .51272) 

=  log  .0631  +  log  7.208  +  log. 51272, 
log   .0631=    8.8000-10 
log   7.208=    0.8578 
log  .51272=    9.7099-10 
Adding,  log  of  result  =  19.3677  -  20 

=    9.3677  -  10  (See  Note  1.) 
Number  corresponding  to  9.3677  —  10  =  .2332. 

Note  1.   If  the  sum  is  a  negative  logarithm,  it  should  be  written  in 
such  a  form  that  the  negative  portion  of  the  characteristic  may  be  — 10. 
Thus,  19.3G77  -20  is  written  in  the  form  9.3677-10. 


866  COLLEGE   ALGEBRA. 

2.  Find  the  value  of  ?^^. 

7984 

By  Art.  500,  log  ?p|  =  log  33G.8  -  log  7984. 

log  336.8  =  12.5273 -10    (See  Note  2.) 
log  7984  =    3.9022 
Subtracting,       log  of  result  =    8.6251  -  10 
Number  corresponding  =  .04218. 

I^ote  2;  To  subtract  a  greater  logarithm  from  a  less,  or  to  subtract. 
a  negative  logarithm  from  a  positive,  increase  the  characteristic  of  the 
mmuend  by  10,  wi-iting  — 10  after  the  mantissa  to  compensate. 

Thus,  to  subtract  3.9022  from  2.5273,  write  the  minuend  in  the  form 
J2.5273  — 10  ;  subtracting  3.9022  from  this,  the  result  is  8.6251  - 10. 

3.  Find  the  value  of  (.07396)^. 

By  Art.  502,     log  (.07396)^  =  5  x  log  .07396. 

log  .07396  =  8.8690  -  10 
5 


44.3450  -  50 
=  4.3450-10  (See  Note  1.) 
=  log. 000002213. 


4.    Find  the  value  of  V. 035063. 

By  Art.  503,    log  ^.035063  =  -  log  .035063. 
o 

log  .035063  =  8.5449  -  10 

3)28.5449  -  30  (See  Note  3.) 

9.5150  - 10 
=  log  .3274. 

Note  3.  To  divide  a  negative  logarithm,  write  it  in  such  a  form 
that  the  negative  portion  of  the  characteristic  may  be  exactly  divisible 
by  the  divisor,  with  —  10  as  the  quotient. 

Thus,  to  divide  8.5449  —  10  by  3,  we  write  the  logarithm  in  the  foiTQ 
28.5449  —  30.     Dividmg  this  by  3,  the  quotient  is  9.5150  -  10. 


LOGARITHMS.  367 


ARITHMETICAL  COMPLEMENT. 

516.  The  Arithmetical  Complement  of  the  logarithm  of  a 
number,  or,  briefly,  the  Cologarithm  of  the  number,  is  the 
logarithm  of  the  reciprocal  of  that  number. 

Thus,     colog  409  =  log  -^  =  log  1  -  log  409. 

log  1  =  10.  -  10  (Note  2,  Art.  515.) 

log  409=    2.6117 
.-.   colog 409=    7.3883-10.. 

Again,  colog  .067  =  log  -^  =  log  1  -  log  .067. 

log  1  =  10.  -  10 

.  log  .067=    8.8261-10 
.-.  colog .067  =    1.1739. 
Tlie  following  rule  is  evident  from  the  above  : 
To  find  the  cologarithm  of  a  number,  subtract  its  logarithm 
from  10  -  10. 

Note.  The  cologarithm  may  be  obtamed  by  subtracting  the  last 
significant  figure  of  the  logarithm  from  10  and  each  of  the  others 
from  9,  — 10  being  written  after  the  result  in  the  case  of  a  positive 
logarithm. 

517.   Example.   Find  the  value  of   — -" 


r09  X  .0946 

log       -5138^        =  log  r.51384  x  ^  X  -^\ 
'^  8.709  X  .0946         ^V  ^-^^^      .0946; 

=  log  .51384  + log -^  + log-   ^ 


8.709         °.0946 
=  log  .51384  +  cotog  8.709  +  colog  .0946. 
log  .51384  =  9.7109 -10 
colog  8.709  =  9.0601 -10 
colog  .0946  =  1.0241 

9.7951  -  10  =  log  .6239. 


368  COLLEGE   ALGEBRA. 

It  is  evident  from  the  above  example  that  the  logaritLiu 
of  a  fraction  is  equal  to  the  logarithm  of  the  numerator  plus 
the  cologarithm  of  the  denominator. 

Or  in  general,  to  find  the  logarithm  of  a  fraction  whose 
terms  are  composed  of  factors, 

Add  together  the  logarithms  of  the  factors  of  the  numerator, 
and  the  cologarithms  of  the  factors  of  the  denominator. 

Note.  The  value  of  the  above  fraction  may  be  found  without  using 
cologarithms,  by  the  following  formula  : 

log 1^1384 ^ ,      .51384  -  log  (8.709  X  .0946) 

^  8. 709  X. 0946         °  o^  ^  ; 

=  log  .51384  -  (log  8. 709  +  log  .0940). 

The  advantage  in  the  use  of  cologarithms  is  that  the  written  work 
of  computation  is  exliibited  in  a  more  compact  form. 


EXAMPLES. 

518.  Note.  A  negative  quantity  has  no  common  logarithm  (Art. 
488,  ]s'ote).  If  such  quantities  occur  in  computation,  they  may  be 
treated  as  if  they  were  positive,  and  the  sign  of  the  result  determined 
irrespective  of  the  logarithmic  work. 

Thus,  in  Ex.  2,  Art.  518,  the  value  of  721.3  X  (-3.0528)  maybe 
obtained  by  finding  the  value  of  721.3  X  3.0528,  and  putting  a  negative 
sign  before  the  result.     See  also  Ex.  24. 

Find  by  logarithms  the  values  of  the  following : 

1.  130.36  X  .08237.  3.    (- 4.32G4)  x  (-.050377). 

2.  721.3  X  (-3.0528).  4.    .27031  x  .042809. 

g    401.8  ^     -.3384  g     15.008  X  (-.0843) 

52.37"  '    .08659 "  "       .06376  x  4.248 

6     '^•^^^1  8     ^-^'^^^  10     (-2563)x  .03442 

10.813'  ■    .64327'  '      714.8  x  (-.511)  * 

J  J  121.6  x  (-9.025) 

(  -  48.3 )  X  3602  X  (  -  .< >856) ' 


LOGARITHMS. 

3G9 

12.    (23.86)1 

15.    lot. 

18.    ^3. 

13.    (-1.0246)'. 

16.    (.8)T. 

19.   v'ioo. 

14.    (.09323)1 

17.    ( -.003186) i 

20.    V.4294. 

21.    V- 

.02305.                22.    V- 

.00005173. 

23.    Find  the 
n.„2^5_ 

1        .  2\/5 
value  of  ■ — —• 

3^ 

-  Ir^n-  9  _J_  Ino-  ^VK  -U  Pnlncr  ^^ 

>.'  (\vt.  mi^ 

1  ^ 

=  log  2  +  -  log  5  +  -  colog  3. 
3  6 

log  2=    .3010 
log  5  =    .6990  ;  divide  by  3  =    .2330 

colog  3  =  9.5229  -  10 ;  multiply  by  ^  :=  9.6024  -  10 


.1364 
=  log  1.369. 


24.    Find  the  value  of    «  - -03296^ 
\     7.962 


""  \  7.962       3         7.962       3 ' 

log  .03296=    8.5180- 

-10 

log   7.962=    0.9010 

3)27.6170- 

-30 

9.2057  - 

-10  = 

=  log  .1606. 
Hesult,  -.160 

Find  the  values  of  the  following : 

25.    2'x3l                   26.    ^. 
4* 

-  (fi)  • 

370  COLLEGE   ALGEBRA 

3 


28. 


/     .08726^3  34.    V.0001289_ 

V      .1321  y    *  A/.0008276' 


41 


29.   ^r.  35. 


(-■7469)« 
-  (.2345)^ 


30.^|.^l-|.  36.     ^^3 


(.08291)^ 


31.    a/2  X  v'lO  X  A/:oi. 
32. 


37. 


el  3258 
\49309' 


V298.54 


49309  38.    (18.9503)"  x(-.l)''. 

31.63V7 


33.  f^^^m 


429    J  39.    V3734.9  X  .00001108. 

40.    (2.6317)^  X  (.71272)1 

,  ,     .,      A/-.008193X  (.06285)^ 
-.98342 

42.    VA)35  X  a/:02667  x  -v/.0072163. 

EXPONENTIAL   EQUATIONS. 

519.  An  Exponential  Equation  is  an  equation  of  the  form 
a'  =  b,  where  x  is  the  unknown  quantity,  and  a  and  b  are 
positive  real  numbers. 

To  solve  an  equation  of  this  form,  take  the  logarithms  of 
l)oth  members ;  the  result  will  be  an  equation  which  can  be 
solved  by  ordinary  algebraic  methods. 

1.    Given  31'  =  23 ;  find  the  value  of  x. 

Taking  the  logarithms  of  both  members, 

log  (310  =  log  23. 

Whence,  x  log  31  =  log  23  (Art.  502) . 

m       f  log  23      1.3(117       (..„..o 

J  lieretore,  ^  =  ;  ~ — r  =  ^r^  r:  =  -91300. 

lu<,'31       1.1914 


LOGARITHMS.  371 

2.    Given  .2'^  =  3 ;  find  the  value  of  x. 
Taking  the  logarithms  of  both  members, 
X  log  .2  =  log  3. 

Whence         x  =  ^-^^=       -^^^^        =    '^^^^    =  -  .6825. 
wnence,        x     ^^^^^      9.3010-10      -.6990 

EXAMPLES. 
Solve  the  following  equations  : 

3.  11^  =  7.  5.   13.18^  =  .0281.  7.   a^  =  &^'c«. 

4.  .3^  =  .8.  6.    .7034^  =  1.096.  8.    ma'' =  7i\ 

9.   21^'-"^  =  9260.        10.    .0513^+^  =  384.4. 

11.  Given  a,  r,  and  I  ;  derive  the  formula  for  n  (Art.  428). 

12.  Given  a,  r,  and  S ;  derive  the  formula  for  n. 

13.  Given  a,  I,  and  S;  derive  the  formula  for  n. 

14.  Given  r,  I,  and  S ;  derive  the  formula  for  n. 

520.    1.  Find  the  logarithm  of  .3  to  the  base  7. 

By  Art.  507, 

W  3  =  l^^Si^  =  M^I^^IlIO  =  _  :5229  ^  _  _g^g^^ 
.        ^'         logio7  .8451  .8451 

EXAMPLES. 

Find  the  values  of  the  following : 

2.  log2l3.  4.   log.365.  6.   log,3  56.31. 

3.  logs  .9.  5.   log.8.0823.  7.   logis  .007228. 

Examples  like  the  above  may  be  solved  by  inspection,  if 
the  number  can  be  expressed  as  an  exact  power  of  the  base. 


372  COLLEGE   ALGEBRA. 

8.  Find  the  value  of  logiB  128. 

Let  logic  128  =  X ;  tlien  16=^  =  128  (Art.  488), 

That  is,  (2^)^  =  2",  or  2^^  =  2^ 

7 
Then,  by  inspection,     4.x*  =  7,  or  x  =  -' 

Therefore,  logi6l28  =  ^- 

9.  Find  the  logarithm  of  243  to  the  base  3. 

10.  Find  the  logarithm  of  7776  to  the  base  36. 

11.  Find  the  logarithm  of  i  to  the  base  27. 

12.  Find  the  logarithm  of  Jj  to  the  base  ^. 

EXPONENTIAL  AND   LOGARITHMIC   SERIES. 
521.   Let  71  be  greater  than  unity. 
By  Art.  288,       [(l  +  1)"]'=  (l  +  i)" 

Expanding  both  members  by  the  Binomial  Theorem,  we 
have 

fl-l-u-Vl  ""(''-^^    1    I  n(n-l)(n-2)-l  7 

L^      71  |2  n'^  •   [3  n'^      J      ■ 

^    ,          1   ,  nxCnx  —  1)     1 
=  l  +  nx---\ ^— ^  •  - 

n  [2  ?r 

"•"  [3  '71-  ^ 

Since,  by  hypothesis,  n  is  greater  than  1,  -  is  numerically 

less  than  1,  and  by  Art.  466  the  series  in  both  members  of 
( 1 )  are  convergent. 


LOGARITHMS.  373 


We  may  write  equation  (1)  in  the  form 


1-1     A-l'ui 


(2) 


J  +  1  +  — +^         K         '^-J     ■ 

x(x )       x(  X  —  -A(  x  —  ~] 

which  holds  however  great  n  may  be. 

Now  let  n  be  indefinitely  increased. 

1  2 

Then  since  the  limit  of  each  of  the  terms  -,  -,  etc.,  is  0 

71  n 
(Art.  210),  the  limiting  value  of  the  first  member  of  (2)  is 

and  the  limiting  value  of  the  second  member  is 

By  the  Theorem  of  Limits  (Art.  213)  these  limits  are 
equal ;  that  is, 

The  series  in  the  second  member  is  convergent  for  every 
value  of  X  (Art.  465)  ;  and  the  series  in  brackets  is  also 
convergent,  for  it  is  obtained  from  the  series  in  the  second 
member  by  putting  1  in  place  of  x. 

Denoting  the  series  in  brackets  by  e,  we  have 
which  holds  for  every  value  of  x. 


374  COLLEGE   ALGEBRA. 

522.  Putting  mx  in  place  of  x  in  (3),  Art.  521, 

e-=l  +  mcc  +  — -  +  —-  +  ....  (4) 

Let  m  =  logs  a,  where  a  is  any  positive  real  number. 
Then  e™  =  a  (Art.  488),  and  e""^  =  a^ 
Substituting  these  values  in  (4),  we  obtain 

a==  =  l+(log,a)a^  +  (log,a)^|+(log,a)3|  +  ...;     (5) 

which  holds  for  all  values  of  x,  and  all  positive  real  values 
of  a. 

The  result  (5)  is  called  the  Exponential  Series. 

523.  The  system  of  logarithms  which  has  e  for  its  base 
is  called  the  Napierian  System,  from  Napier,  the  inventor 
of  logarithms. 

The  approximate  value  of  e  may  be  readily  calculated 
from  the  series  of  Art.  521, 

and  will  be  found  to  equal  2.7182818... . 

524.  To  expand  log<,(l  +  x)  in  ascending  piowers  of  x. 

Substituting  in  (5),  Art.  522,  1  +  a;  in  place  of  a,  and  y  in 
place  of  X,  we  have 

{l-\-xy  =  l-\-  [log<,(l  +  x)  ]  ?/  +  terms  in  y^-,  y^,  etc. ; 

which  holds  for  all  values  of  y,  and  all  real  values  of  x  alge- 
braically greater  than  —1. 

Expanding  the  first  member  by  the  Binomial  Theorem, 

[2  |3_ 

=  1  +  [log,(l  -{-x)^y+  terms  in  y-,  f,  etc.      (6) 


LOGARITHMS.  375 

The  first  member  of  (6)  is  convergent  when  x  is  numeri- 
cally less  than  1  (Art.  466). 

Hence,  equation  (6)  holds  for  all  values  of  y,  and  for  all 
real  values  of  x  numerically  less  than  1. 

Then,  by  the  Theorem  of  Undetermined  Coefficients,  the 
coefficients  of  y  in  the  two  series  are  equal ;  that  is, 

^ -!4^' +  i  ^-^  -  [t  •'^' +  -  =  ^''^^  (^  +  ^')- 

l£         iZ         II 
Or,        log:(l+a;)  =  x-|  +  f-f  +  |--;  (7) 

which  holds  for  all  real  values  of  x  numerically  less  than  1. 
(Compare  Art.  464.) 

This  result  is  called  the  LogaritJimic  Series. 

Putting  —  a;  in  place  of  x  in  (7),  we  obtain 

log,(l-x)  =  -._f-f-f-f-....         (8) 

Formula  (7)  can  be  used  for  the  calculation  of  Napierian 
logarithms,  provided  x  is  taken  numerically  less  than  1 ;  but 
unless  x  is  small,  it  requires  the  sum  of  a  great  number  of 
terms  to  ensure  any  degree  of  accuracy. 

525.  To  derive  a  more  convenient  formula  for  the  calcida- 
tion  of  Napierian  logarithms. 

Subtracting  equation  (8),  Art.  524,  from  (7),  we  ha.ve 

log,  (1  +  «)  -  log,(l  -x)=^2x-{-~^^~  +  -- 

o  o 

■  Whence,  by  Art.  600, 


1  —  x       V       ^      y 

m. 


1  + 
Y    ,  m-n     ..       1  +  X.  m  +  n      2  m  _  m 

Let  x  = :  then  -— ^ —  = =  ---  =  — 

m  +  ?i  1  — X-      i_'!!L:lVi      ^'*'       '>^ 

m  +  n 


376  COLLEGE   ALGEBRA. 

Substituting  tliese  values  iu  (9),  we  obtain 


n  m  +  n 


1  I'm  —  n\^      1  I'm  —  nV  ~| 

3  \iii  +  nj       5  \vi  +  nj  J 


But  by  Art.  498,  log^—  =  log.m  —  log,,  71 ;  whence, 


log.m : 


log,n  +  2   — -  +  0   —     +i   •— 7-    +•••   • 

[_m  -\-n      6  \m  +  nj       5  \m + nJ  J 


526.  Let  it  be  required,  for  example,  to  calculate  the 
Napierian  logarithm  of  2  to  six  places  of  decimals. 

Putting,  m  =  2  and  n=l  in  the  result  of  Art.  525,  we  have 

log.2  =  log.l+2ri  +  lflY+^riY+-T  :^      . 

Or  since  log.l  =  0  (Art.  494),       '  'C   ' jf   ,  "  /.  -g  T  6  '^'  '7  ^^ 
log,2  =  2  (.3333333  +  .0123457  +  .0008230  +  .0000653 
+  .0000056  +  .0000005  +••••) 
=  2  X  .3465734  =  .6931468 
=  .693147,  to  the  nearest  sixth  decimal  place. 
Having  found  log^2,  we  may  calculate  loge3  by  putting 
m  =  2)  and  n  =  2  in  the  result  of  Art.  525. 

Proceeding  in  this  way,  we  shall  find  log^lO  =  2.302585... . 

527.  To  calculate  the  common  logarithm  of  a  mimber,  hav- 
ing given  its  Napierian  logarithm. 

Putting  6  =  10  and  a  =  e  in  the  result  of  Art.  507, 

1  log,  m  1  T  ,n,  o<  1  I  r        1 

logio  «i  =  ,    ^%,,  =  ^  onorcT  ^  ^^S. ^n  =  .4o42945  X  log, VI. 
logelO      2.o02i^b5 

For  example, 

logi,j2  =  .4342945  x  .693147  =  .301030. 

528.  Tlie  multiplier  by  which  logarithms  of  any  system 
are  derived  from  Napierian  logarithms,  is  called  the  modulus 
of  that  system. 


LOGARITHMS.  377 

Thus,  .4342945  is  the  modulus  of  the  common  system. 

Conversely,  to  find  the  jSTapierigin  logarithm  of  a  number 
when  its  common  logarithm  is  given,  we  may  either  divide 
the  common  logarithm  by  the  modulus  .4342945,  or  multi- 
ply it  by  2.302585,  the  reciprocal  of  .4342945. 

Note.  Napierian  logarithms  are  sometimes  called  hyperbolic  loga- 
rithms, from  having  been  originally  derived  from  the  hyperbola.  They 
are  also  sometimes  called  natural  logarithms,  from  being  those  which 
occur  first  m  the  investigation  of  a  method  of  calculating  logarithms. 

MISCELLANEOUS     EXAMPLES. 

529.  Using  the  table  of  common  logarithms,  find  the 
Napierian  logarithm  of  each  of  the  following  to  foiu-  signi- 
ficant figures  : 

1.  100.  3.^  88.2.  5.    .3437. 

2.  .00001.  4.   1325.  6.    .085623. 
(7y  What  is  the  characteristic  of  log3400  (Art.  488)  ?  ^'  ■ 

8.    AVhat  is  the  characteristic  of  logo  1523? 

(9,    If  log  2  =  .3010,  how  many  digits  are  there  in  2'*^  ? 

(\Q,    If  log  6  =  .7782,  how  many  digits  are  there  in  6-^  ? 

11.  If  log  7  =  .8451,  how  many  digits  are  there  in  the 
integral  part  of  7"8  ? 


378  COLLEGE   ALGEBRA. 


XXXV.    COMPOUND   INTEREST  AND 
ANNUITIES. 

530.  The  principles  of  logarithms  may  be  applied  to  tlie 
solution  of  problems  in  Compound  Interest. 

Let  P  =  the  principal  in  dollars ; 
n  —  the  number  of  years  ; 

t  =  the  ratio  to  one  year  of  the  time  during  which 
simple   interest   is    calculated ;    for  instance, 
if  the  interest  is  compounded  semi-annually, 
t  =  \\ 
R  =  the  amount  of  one  dollar  for  the  time  t ; 
A  =  the  amount  of  P  dollars  for  n  years. 

1 .    Given  P,  n,  t,  R  ;  to  find  A. 

Since  the  amount  of  one  dollar  for  the  time  t  is  R,  the 
amount  of  P  dollars  for  the  same  period  will  be  PR. 

That  is,  the  amount  at  the  end  of  the  1st  interval  is  PR. 
In  like  manner,  the  amount  at  the  end  of  the 

2d  interval  =  PR  x  R  =  PR" ; 

3d  interval  =  PR'  x  R  =  PR^ ;  etc. 

Since  the  whole  number  of  intervals  is  -,  the  amount  at 

t 

the  end  of  the  last  one,  in  accordance  with  the  law  observed 

above,  will  be  PR'. 

That  is,  A  =  PR''.  (1) 

By  logarithms,  log  A  =  log  P  +  "  log  R.  (2) 

Example.  What  will  be  the  amount  of  $  7326  for  3  years 
and  9  months  at  7  per  cent  compound  interest,  the  interest 
being  compounded  quarterly  ? 


COMPOUND   INTEREST   AND   ANNUITIES.        379 

111  this  case, 

P=  7326,  n  =  3f,  t  =  \,  R  =  1.0175,  -=15. 

log  P=  3.8649 
log  R  =  0.0075;  multiply  by  15  =  0.1125 

log  ^  =  3.9774 
.-.  .4  =  19492. 

2.  Given  n,  t,  B,  A;  to  find  P. 

Prom  (2),  log P  =  log A-- log R. 

Example.   What  sum  of  money  will  amount  to  $  1763.50 
in  3  years  at  5  per  cent  compound  interest,  the  interest 
being  compounded  semi-annually  ? 
In  this  case, 

n  =  3,t  =  ^,R  =  1.025,  A  =  1763.5,  ""  =  6. 
log  .4  =  3.2464 
log  R  =  0.0107 ;  multiply  by  6  =  0.0642 
log  P=  3.1822 
.-.  P=  $1521.40. 

3.  Given  P,  t,  R,  A;  to  find  n. 

From  (2),     -  log  i?  =  log  ^1  -  log  P. 

Whence,  ^^  ^  ^log^  -  log  P). 

logP 

Exam'ple.  In  how  many  years  will  $300  amount  to 
$  396.90  at  6  per  cent  compound  interest,  the  interest  being 
compounded  quarterly  ? 

Here,      P  =  300,  t  =  \,  P  =  1.015,  ^  =  396.9. 

-,^  -  lo^^  396-9  -  log  300  ^  2.5987  -  2.4771  ^  .1216 
4  log  1.015  4  X. 0064  .0256 

=  4.75  years. 


380                             COLLEGE    ALGEBRA. 
4.    Given  P,  n,  t,  A;  to  find  R. 
From  (2),  log  72  =  —1:: 2 — 

7 
Example.   At  what  rate  per  cent  per  annum  will  f  500 
amount  to  $  688.83  in  6  years  and  6  months,  the  interest 
being  compounded  semi-annually  ? 

Here,  P=500,  « =  61    t  =  \,  ^  =  688.83,  -  =  13. 

log^=       2.8381 
lotr  P  =       2.6990 


13)0.1391 
log  R  =  0.0107 
.:  R=^       1.025. 

That  is,  the  interest  on  one  dollar  for  0  months  is  $  .025, 
and  the  rate  is  5  per  cent  per  annum. 

531.  To  find  the  present  worth  of  A  dollars  due  at  the  end 
of  n  years,  the  interest  being  compounded  annually. 

I'utting  t  —  1  in  (1),  Art.  530,  we  have 

A  =  PR" ;  whence,  P  =  ^• 

ANNUITIES. 

532,  An  Annuity  is  a  fixed  sum  of  money  payable  at 
equal  intervals  of  time. 

In  the  present  chapter  we  shall  consider  those  cases  only 
in  which  the  payments  are  annual ;  in  finding  th'e  present 
worth  of  such  an  annuity,  it  is  customary  to  compound  the 
interest  annually;  and  when  we  speak  of  the  annuity  as 
beginning  at  a  certain  epoch,  it  is  understood  that  the  first 
payment  becomes  due  one  year  from  that  time. 


COMPOUND   INTEREST   AND   ANNUITIES.         381 

533.    To  find  the  present  worth  of  an  annuity  to  continue 
for  n  successive  years,  alloiving  comjJound  interest. 

Let        A  =  the  annuity  in  dollars  ; 

Ji  =  the  amount  of  one  dollar  for  one  year ; 
P  =  the  present  worth  of  the  annuity. 

By  Art.  531,  the  present  worth  of  the 

1st  payment  =  —  ; 
R 

2d    payment  =  —  ; 


nth  payment  = 

Hence  the  sum  of   the   present  worths  of   the  separate 
payments,  or  the  present  worth  of  the  annuity,  is 

R-  ^  i2»-i  ^  B'     R 

Thatis,     P=^^l  +  ^^  +  ...+-l+lj. 

The  expression  in  brackets  is  the  sum  of  the  terms  of  a 
ometrical  progression,  in  w 

Whence  by  (II.),  Art.  426, 


geometrical  progression,  in  which  a  =  — ,  r  =  R,  and  I  =  -- 


i-i 

Example.  Find  the  present  worth  of  an  annuity  of  $  150 
to  continue  for  20  years,  allowing  4  per  cent  compound 
interest. 

Here,  ^  =  150,  ?i  ^  20,  ^  =  1.04,  i?  -  1  =  .04. 


Whence,  P  =  IJi' [l  - -_^,]. 


382  COLLEGE    ALGEBRA. 

colog  1.04  =  9.9830 

20 

9.6600 

Number  correspondiug  =    .4571. 

Therefore,  ^=  i??  (1-.4571)  =3750  x  .5429. 

.04 

log  3750  =  3.5740 
log  .5429  =  9.7347 

log  P=  3.3087 

.-.  P=  $2035.70. 


534.   We  have  from  (3),  Art.  533, 

P(R-l)      PB'^(R-l) 

J2" 


(4) 


whieh  is  a  formula  for  finding  the  annuity  to  continue  for  n 
successive  years,  when  the  present  worth  and  the  amount  of 
one  dollar  for  one  year  are  given. 

Note.  Formula  (4)  may  also  be  used  to  find  what  fixed  annual 
payment  must  be  made  to  cancel  a  note  of  P  dollars  due  n  years  hence, 
E  being  the  amount  of  one  dollar  for  oiie  year. 

535.  If  in  (3),  Art.  533,  n  is  indefinitely  increased,  the 
limiting  value  of  the  second  member  is 

^     (Art.  210). 


E-1 


That  is,  theiiresent  loorth  of  a  perpeUml  anmiity  is  equal  to 
the  amount  of  the  annuity  divided  by  the  interest  on  one  dollar 
for  one  year. 


COMPOUND   INTEREST   AND   ANNUITIES.       383 

636.  To  find  the  present  worth  of  an  annuity  to  begin  after 
m  years  and  then  continue  for  n  years,  alloiving  compound 
interest. 

"Witli  the  notation  of  Art.  533,  the  value  of  the  annuity 
one  year  before  the  first  payment  becomes  due,  is 

B-1'        B''(R-1) 

By  Art.  531,  the  present  worth  of  the  above  amount,  due 
m  years  hence,  is 

A(R'>~1)  .  ^, 

Therefore,  P  =    M^"-'^)  . 

537.  By  Art.  535,  the  present  worth  of  a  perpetual  annu- 
ity to  begin  after  m  years,  is  given  by  the  formula 

p^—A 

R'"{E-l) 

EXAMPLES. 

538.  1.  What  will  be  the  amount  of  $1000  for  18  years 
at  6  per  cent  compound  interest,  the  interest  being  com- 
pounded annually  ? 

2.  What  sum  of  money  will  amount  to  f  870.50  in  7  years 
and  3  months  at  3  per  cent  compound  interest,  the  interest 
being  compounded  quarterly  ? 

3.  In  how  many  years  will  $968  amount  to  $1209.40  at 
5  per  cent  compound  interest,  the  interest  being  compounded 
semi-annually  ? 

4.  What  is  the  present  worth  of  a  note  for  f  514.23  due 
11  years  hence,  allowing  41  per  cent  compound  interest,  the 
interest  being  compounded  annually  ? 


384  COLLECxE  ALGEBRA. 

5.  At  what  rate  per  cent  per  annum  will  f  2600  gain 
$416.40  in  3  years  and  9  months,  the  interest  being  com- 
pounded quarterly  ? 

6.  In  how  many  years  will  a  sum  of  money  double  itself 
at  5  per  cent  compound  interest,  the  interest  being  com- 
pounded annually  ? 

7.  In  how  many  years  will  a  sum  of  money  treble  itself  at 
7  per  cent  compound  interest,  the  interest  being  compounded 
semi-annually  ? 

8.  What  sum  of  money  will  amount  to  $1000  in  11 
years  and  8  months  at  3f  per  cent  compound  interest,  the 
interest  being  compounded  every  four  months  ? 

9.  What  is  the  present  worth  of  an  annuity  of  $200  to 
continue  15  years,  allowing  5  per  cent  compound  interest  ? 

10.  What  is  the  present  worth  of  an  annuity  of  $  1127  to 
continue  3  years,  allowing  7  per  cent  compound  interest  ? 

11.  What  is  the  present  worth  of  an  annuity  of  $1570  to 
begin  after  11  years  and  continue  for  6  years,  allowing  4  per 
cent  compound  interest  ? 

12.  What  fixed  annual  payment  must  be  made  in  order 
to  cancel  a  note  for  $  2000  in  7  years,  allowing  3^  per  cent 
compound  interest  ? 

13.  What  is  the  present  worth  of  a  perpetual  annuity  of 
$  186.25,  to  begin  after  7  years,  allowing  3i  per  cent  com- 
pound interest  ? 

14.  What  annuity  to  continue  10  j'ears  can  be  purchased 
for  $2038,  allowing  6  per  cent  compound  interest  ? 

15.  A  person  borrows  $  525,4  ;  how  much  must  he  pay  in 
annual  instalments  in  order  that  the  whole  debt  may  be  dis- 
charged in  12  years,  allowing  41-  per  cent  compound  interest? 


PERMUTATIONS   AND   COMBINATIONS.  385 


XXXVI.     PERMUTATIONS  AND   COM- 
BINATIONS. 

539.  The  different  orders  in  which  things  can  be  arranged 
are  called  their  Permutation?.  >u  ^  ^^ 

Thus,  the  perniutatioliS  of  the  letters  a,  b,  c,  taken  two  at 
a  time,  are  ab,  ac,  ba,  be,  ca,  cb;  and  their  permutations 
taken  three  at  a  time,  are  abc,  acb,  bac,  bca,  cab,  cba. 

540.  The  Combinations  of  things  are  the  different  collec- 
tions which  can  be  formed  from  them,  without  regard  to. 
the  order  in  which  they  are  placed. 

Thus,  the  combinations  of  the  letters  a,  b,  c,  taken  two  at 
a  time,  are  ab,  be,  ca ;  for  though  ab  and  ba  are  different 
permutations,  they  form  the  same  combination. 

541.  To  find  the  number  of  permutations  of  n  different 
things  taken  two  at  a  time. 

Consider  the  letters  a^,  a^,  O3,  a^,  ...,  a„. 

The  permutations  of  the  letters  taken  two  at  a  time,  hav- 
ing tti  as  the  first  element,  are 

a^a^,  ctiOg,  a^tti,  ...,  aia^\ 

the  number  of  which  is  n  —  1. 

In  like  manner,  there  are  n  —  1  different  permutations  of 
the  letters  taken  two  at  a  time,  having  ag  as  the  first 
element ;  and  similarly  for  each  of  the  remaining  letters, 
«3,  a^,  ...,  a„. 

Therefore  the  whole  number  of  permiitations  of  the  let- 
ters taken  two  at  a  time  is  equal  to 

n{n  —  1). 


386  COLLEGE   ALGEBRA. 

542.    We  will  now  discuss  the  general  case. 

To  find  the  number  of  permutations  of  n  different  things 
taken  r  at  a  time. 

Consider  the  letters 

ffli,  a.2,  ttg,  ...,  a„  a,+i,  a,+2,  •••?  «„■ 

Take  any  permutation  containing  r  letters,  for  example 
the  one  consisting  of  the  first  r  letters  in  their  order  ;  that 
is, 

Oittoag ...  a,. 

Placing  after  this  the  remaining  n  —  r  letters  one  at  a 
time,  in  the  following  manner, 

ajajC^  •  •  •  ttr^r+l 
aia^ag . . .  a^a^+2 


OiOotta . . .  a^a„ 


there  are  formed  n  —  r  different  permutations,  each  contain- 
ing r -f  1  letters. 

We  may  proceed  in  a  similar  manner  with  the  other  per- 
mutations containing  r  letters,  and  in  each  case  obtain  n  —  r 
different  permutations,  each  containing  r  +  1  letters. 

It  is  evident  that  the  permutations  of  r  +  1  letters  formed 
in  the  above  manner  are  all  different ;  also,  that  we  obtain 
in  this  way  all  the  permutations  containing  r  + 1  letters. 

Hence,  the  number  of  permutations  of  the  letters  taken  r 
at  a  time,  multiplied  by  w  —  r,  is  equal  to  the  number  of 
permutations  of  the  letters  taken  r  + 1  at  a  time. 

But  the  number  of  permutations  of  the  letters  taken  two 
At  a  time,  is  equal  to  n{n  —  1)  (Art.  541). 

Hence  the  number  of  permutations  of  the  letters  taken 
three  at  a  time,  is  equal  to  the  number  taken  two  at  a  time, 
multiplied  by  n  —  2,  or  n(n  —  1)  (n  —  2). 


PERMUTATIONS   AND   COMBINATIONS.  387 

The  number  of  permvitations  of  the  letters  taken  four  at 
a  time,  is  equal  to  the  number  taken  three  at  a  time,  multi- 
plied by  n  —  3,  or  ?i(h  —  1)  (?i  —  2)  {n  —  3)  ;  and  so  on. 

It  will  be  observed  that  the  last  factor  in  the  number  of 
permutations  is  n,  minus  a  number  one  less  than  the  num- 
ber of  letters  taken  at  a  time. 

Hence  the  number  of  permutations  of  the  letters  taken  r 
at  a  time  is  n{ii  —  1)  {n  —  2)"-[n  —  (?•  —  1)]  ;  that  is, 

„P,  =  n{n  -  1)  (n  -  2)--{n  -  r  +  1).  (l^) 

Note.  The  number  of  permutations  of  n  different  tilings  taken  r  at 
a  time,  is  usually  denoted  by  the  symbol  „Pr. 

543.  If  all  the  letters  are  taken  together,  r  =  n,  and 
formula  (1)  becomes 

„P„  =  n{n  -  1)  (w  -  2) ... 3 . 2  •  1  =[n.  (2) 

That  is,  tlie  number  of  penmitations  of  n  different  thuigs 
taken  n  at  a  time  is  equal  to  the  product  of  the  natural  num- 
bers from  1  to  n  inclusive.     (See  Note,  Art.  444.) 

544.  To  find  the  number  of  combinations  of  n  different 
things  taken  r  at  a  time. 

The  number  of  permxdations  of  n  different  things  taken  r 
at  a  time,  is 

n{n-l){n-2)--{n-r  +  l)  (Art.  542). 

But  by  Art.  543,  each  combination  of  r  different  things 
may  have  [r  permutations. 

Hence  the  number  of  combinations  of  n  different  things 
taken  r  at  a  time  is  equal  to  the  number  of  permutations, 
divided  by  [r;  that  is, 

(7,  ==  n(n-l){n-2):.{n-r  +  l)  ^^^ 

Note.  The  number  of  combinations  of  n  different  things  taken  r  at 
at  a  time,  is  usually  denoted  by  tlie  symbol  ^C,. 


388  COLLEGE   ALGEBRA. 

545.  Multiplying  both  terms  of  the  second  member  of 
(3)  by  the  product  of  the  natural  numbers  from  1  to  n  —  r 
inclusive,  we  have 


,a  = 


7i{n-l)  •••  {n-r  +  1)-  (?t  -  r)  •••  2  •  1  _ 


|r  X  l-2...(n-;-)  \r\n-r' 

which  is  another  form  of  the  result. 

546.  By  Art.  545,  the  number  of  combinations  of  n  differ- 
ent things  taken  n  —  r  at  a  time,  is 

\n  \n 

— ,  or  — 

\n  —  r\n  —  {n  —  r)  |  n  —  r  \  r 

But  this  is  the  same  as  the  number  of  combinations  of  n 
different  things  taken  r  at  a  time. 

Hence,  the  number  of  combinations  of  n  different  things 
taken  r  at  a  time  is  equal  to  the  number  of  combinations  of 
n  different  things  taken  n  —  r  at  a  time. 

EXAMPLES. 

547.  1-  HoAV  many  changes  can  be  rung  with  10  bells, 
taking  7  at  a  time  ? 

Putting  n  —  10  and  r  =  7  in  (1),  Art.  542, 

loPr  =  10 . 9  •  8  •  7  •  G  •  5  •  4  =  G04800. 

2.  How  many  different  combinations  can  be  formed  with 
16  letters,  taking  12  at  a  time  ? 

By  Art.  546,  the  number  of  combinations  of  16  different 
things  taken  12  at  a  time  is  equal  to  the  number  of  com- 
l)inations  of  16  different  things  taken  4  at  a  time. 

I'utting  n  =  16  and  r  =  4  in  (3),  Art.  544,  we  have 
^_16.15.14.13_ 


PERMUTATIONS   AND   COMBINATIONS.         389 

Find  the  values  of  the  following : 

3.  lyPg.  5.       31 P,.  7.       I8C12. 

4.  gXg.  b.      1465.  O.      22^15' 

9.    How  many  permutations  can  be  formed  from  the  26 
letters  of  the  alphabet,  taken  5  at  a  time  ? 

10.  How  many  different  words  of  seven  letters  each  can 
be  formed  from  the  letters  in  the  word  forming  ? 

11.  How  many  different  numbers,  of  6  different  figures 
each,  can  be  formed  from  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  9  ? 

12.  From  a  company  of  50  soldiers,  how  many  different 
pickets  of  6  men  can  be  takgn  ? 

13.  How  many  different  words  of  4  letters  each  can  be 
made  with  7  letters  ?  How  many  of  3  letters  each  ?  How 
many  of  6  letters  each  ?     How  many  in  all  possible  ways  ? 

14.  How  many  different  committees  of  12  persons  each 
can  be  formed  out  of  a  corporation  of  20  persons  ? 

15.  There  are  12  points  in  a  plane,  of  which  no  three  are 
in  the  same  straight  line.  How  many  different  triangles 
can  be  formed,  having  three  of  the  points  for  vertices  ? 

16.  How  many  different  numbers  of  5  different  figures 
each,  can  be  formed  from  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  9,  0  ? 

17.  How  many  different  throws  can  be  made  with  two 
dice  ? 

18.  How  many  different  throws  can  be  made  with  three 
dice  ? 

19.  How  many  different  words,  each  consisting  of  4 
consonants  and  2  vowels,  can  be  formed  from  8  consonants 
and  4  vowels  ? 

The  number  of  combinations  of  the  8  consonants,  taken  4 

.    8  •  7  •  6  •  5 
at  a  time,  is  - — ; — -; — ,  or  70. 
1  •  2  •  3  •  4 


390  collegp:  algebra. 

Tlie  number  of  combinations  of  the  4  vowels,  taken  2  at  a 
time,  IS  - — -,  or  6. 

Any  one  of  the  70  sets  of  consonants  may  be  associated 
with  any  one  of  the  6  sets  of  vowels ;  hence,  there  are  in  all 
70  X  6,  or  420  sets,  each  containing  4  consonants  and  2  vowels. 

But  each  of  the  sets  of  6  letters  may  have  [6,  or  720  dif- 
ferent permutations  (Art.  543). 

Therefore  the  whole  number  of  different  words  is 
420  X  720,  or  302400. 

20.  How  many  different  words,  each  consisting  of  3  con- 
sonants and  1  vowel,  can  be  formed  from  12  consonants  and 
3  vowels? 

21.  How  many  different  committees,  each  consisting  of  2 
Kepublicans  and  3  Democrats,  can  be  formed  from  14 
Kepublicans  and  21  Democrats  ? 

22.  Out  of  9  red  balls,  4  Avhite  balls,  and  6  black  balls, 
how  many  different  combinations  can  be  formed,  each  con- 
sisting of  5  red  balls,  1  white  ball,  and  3  black  balls  ? 

23.  How  many  different  words,  each  consisting  of  4  con- 
sonants and  3  vowels,  can  be  formed  from  10  consonants 
and  5  vowels  ? 

24.  How  many  different  words,  each  consisting  of  4  con- 
sonants and  1  vowel,  can  be  formed  from  10  consonants  and 
3  vowels,  the  vowel  being  the  middle  letter  of  each  word  ? 

25.  How  many  different  words  of  8  letters  each  can  be 
formed  from  4  consonants  and  4  vowels,  the  vowels  always 
occupying  the  even  places  ? 

26.  Out  of  11  physicians,  13  teachers,  and  8  lawyers, 
how  many  different  committees  can  be  formed,  each  consist- 
ing of  3  physicians,  4  teachers,  and  2  lawyers? 


PERMUTATIONS   AND   COMBINATIONS.  391 

27 o  How  many  different  words  of  7  letters  each  can  be 
formed  from  the  letters  a,  b,  c,  d,  e,  f,  g,  each  word  being 
such  that  the  letters  a,  b,  c  are  never  separated  ? 

28.  How  many  different  words  of  5  letters  each  can  be 
formed  from  the  letters  in  the  word  Cambridge,  each  word 
beginning  with  a  vowel,  and  ending  with  a  consonant,  and 
having  a  consonant  for  its  middle  letter  ? 

29.  In  how  many  different  ways  can  52  cards  be  arranged 
in  four  sets,  each  set  containing  13  cards  ? 

548.  To  find  the  number  of  permutations  of  n  things  loliich 
are  not  all  different,  taken  all  together. 

Let  there  be  w  letters,  of  which  p  are  a's,  q  are  6's,  and  r 
are  c's,  the  rest  being  all  different. 

Let  JV  denote  the  number  of  permutations  of  these  letters 
taken  all  together. 

If,  in  any  assigned  permutation  of  the  7i  letters,  the  p  a's 
were  replaced  by  p  new  letters,  differing  from  each  other 
and  also  from  the  remaining  n—p  letters,  then  by  simply 
altering  the  order  of  these  p  letters  among  themselves,  with- 
out changing  the  positions  of  any  of  the  other  letters,  we 
could  from  the  original  permutation  form  \_p  different  per- 
mutations (Art.  543). 

If  this  were  done  in  the  case  of  each  of  the  N  original 
permutations,  the  whole  number  of  permutations  would  be 

Again,  if  in  any  one  of  the  latter  the  q  6's  were  replaced 
by  q  new  letters,  differing  from  each  other  and  from  the 
remaining  n—q  letters,  then  by  altering  the  order  of  these 
q  letters  among  themselves,  we  could  from  the  original  per- 
mutation form  [g  different  permutations ;  and  if  this  were 
done  in  the  case  of  each  of  the  N  x\_p  original  permutations, 
the  whole  number,  of  permutations  would  he  N  x[px[2- 


392  COLLEGE   ALCxEBRA. 

In  like  manner,  if  in  each  of  the  latter  the  r  c's  were 
replaced  by  r  new  letters,  differing  from  each  other  and 
from  the  remaining  oi  —  r  letters,  and  these  r  letters  were 
permuted  among  themselves,  the  whole  number  of  permu- 
tations would  heNx[px[qx\r. 

But  the  number  of  permutations  on  the  liypothesis  that 
the  n  letters  are  all  different,  is  [n  (Art.  543). 

Therefore,  N x\px\q  x\r  =  \n  ;  or,  iV=  — — — 
Any  other  case  may  be  treated  in  a  similar  manner. 


EXAMPLES. 

1 .  How  many  different  permutations  can  be  formed  from 
the  letters  of  the  word  Tennessee,  taken  all  together  ? 

2.  How  many  different  words  of  twelve  letters  each  can 
be  formed  from  the  letters  in  the  word  independence  9 

3.  In  how  many  ways  can  6  dimes,  4  quarter-dollars,  and 
3  half-dollars  be  distributed  among  13  boys,  so  that  each 
may  receive  a  coin  ? 

4.  In  how  many  ways  can  15  balls  be  arranged  in  a  row, 
if  7  of  the  balls  are  white,  5  black,  and  3  red? 

5.  How  many  different  words  of  seven  letters  each  can  be 
formed  from  the  letters  in  the  word  Algebra,  the  first,  fourth, 
and  last  letters  being  vowels  ? 

6.  How  many  different  numbers  of  seven  figures  each 
can  be  formed  from  the  digits  1,  2,  3,  4,  3,  2,  1,  the  first, 
third,  fifth,  and  last  digits  being  odd  numbers  ? 


PROBABILITY. 


XXXVII.    PROBABILITY  (CHANCE). 

549.  Definition.  If  an  event  can  happen  in  a  ways,  and 
fail  to  happen  in  h  ways,  and  all  these  ways  are  equally 
likely  to  occnr,  the  probability  or  chcuice  of  the  happening  of 

the  event  is ,  and  the  probability  of  its  failing  is 

a+b  a-j-b 

Or,  we  say  that  the  odds  are  a  to  b  in  favor  of  the  event, 

if  a  is  greater  than  b,  and  a  to  &  against  the  event,  if  a  is 

less  than  b. 

550.  It  follows  that  if  the  probability  of  the  happening 
of  an  event  is  p,  the  probability  of  its  failing  is  1  —p. 

551.  If  an  event  is  certain  to  happen,  b  is  equal  to  zero, 
and  the  probability  of  the  happening  of  the  event  is  -,  or  1. 

552.  Example  .1.  A  bag  contains  5  red  >  balls,  4  white 
balls,  and  3  black  balls. 

(a.)  If  one  ball  is  drawn,  what  is  the  probability  that  it 
is  white  ? 

The  drawing  of  a  white  ball  can  happen  in  4  ways,  since 
either  one  of  the  4  white  balls  may  be  drawn.  It  can  fail 
to  happen  in  8  ways,  since  either  one  of  the  red  or  black 
balls  may  be  drawn. 

Hence,  the  probability  of  drawing  a  white  ball  is -, 

or  -• 
3 

(6.)    If  3  balls  are  drawn,  what  is  the  probability  that 

they  are  all  red  ? 

The  number  of  combinations  of  the  5  red  balls,  taken  3 

K    4    o 

at  a  time,  is  '  ' /]   (Art.  544),  or  10;  that  is,  the  drawing 
of  3  red  balls  can  happen  in  10  ways. 


394  COLLEGE   ALGEBRA. 


time,  is  12jjl^^  ^^.  goQ ;  that  is,  the  drawing  of  3  balls 
1  •  U  •  3 


The  number  of  combinations  of  the  12  balls,  taken  3  at 

,.         .    12.11.10 

a  time,  is  ,  oi 

1-2.3 

can  occur  in  220  ways. 

Hence,  the  ijrobability  of  drawing  3  red  balls  is ,  or  — 

220       22 

(c.)  If  6  balls  are  drawn,  what  is  the  probability  that  2 
are  red,  3  white,  and  1  black  ? 

The  number  of  combinations  of  the  5  red  balls,  taken  2  at 

5-4 

a  time,  is ,  or  10 ;  the  number  of  combinations  of  the  4 

1*2  4.3.2 

white  balls,  taken  3  at  a  time,  is  ,  or  4. 

1.2.3 

We  may  associate  together  any  one  of  the  10  combina- 
tions of  red  balls,  any  one  of  the  4  combinations  of  white 
balls,  and  any  one  of  the  3  black  balls ;  hence  there  are  in 
all  10  X  4  X  3,  or  120  combinations,  each  consisting  of  2  red 
balls,  3  white  balls,  and  1  black  ball. 

Also,  the  number  of  combinations  of  the  12  balls,  taken  6 

^     ^.         .    12.11.10.9.8.7        no< 

at  a  time,  is ,  or  924. 

1.2.3.4.5.6    ' 

Hence,  the  required  probability  is  -^^,  or  — . 

Exawx)le  2.  A  bag  contains  30  tickets  numbered  1,  2,  3, 
...,  30. 

(a.)  If  four  tickets  are  drawn,  what  is  the  chance  that 
hoth  1  and  2  will  be  among  them  ? 

Tlie  number  of  combinations  of  the  28  tickets  numbered 

98 .  97 
3,  4,  ...,  30,  taken  2  at  a  time,  is  - — j^;  that  is,  there  are 
00  ^  97 .  . 

— '-^^  ways  of  drawing  four  tickets,  two  of  -^vhich  are  num- 
1.2         -^ 

bered  1  and  2. 

The  number  of  combinations  of  the  30  tickets,  taken  4  at 

.30.29.28.27 
a  time,  is      ^^  ^  ^  ^ — 


PROBABILITY.  395 

Hence  the  probability  tliat,  if  four  tickets  are  drawn,  t^ro 
of  them  will  be^l  and  2,  is 

28 •  27 ^ .30 •  29 •  28 •  27  _    3.4    _ _^ 
1.2     ■       1.2.3-4      ~30.29"~145' 
{b.)    If  four  tickets  are  drawn,  what  is  the  chance  that 
eitlier  1  or  2  will  be  among  them  ? 

Either  1  or  2  will  be  among  the  tickets  drawn,  unless 
each  of  the  four  bears  a  number  from  3  to  30  inclusive. 
The  number  of  combinations  of  the  28  tickets  numbered 

90     97    9fj    OK 

3,  4,  ...,  30,  taken  4  at  a  time,  is  ^    ^^^  -^o-^^. 

The  number  of  combinations  of  the  30  tickets,  taken  4  at 
30.29.28.27 


a  time,  is 


1-2.3.4 


Hence  the  probability  that  each  of  the  4  tickets  drawn 

.  1      .        o^    on-     1     •       •   28.27-26.25        65 

bears  a  number  from  o  to  30  inckisive,  is ,  or  — 

'      30 .  29  •  28  •  27        87 

Therefore  the  probability  that  each  of  the  4  tickets  drawn 

65 

does  not  bear  a  number  from  3  to  30  inclusive,  is  1 

99  87 

(Art.  550),  or  ^. 

This  then  is  the  probability  that  either  1  or  3  will  be 
among  the  tickets  drawn. 


EXAMPLES. 

553.  1.  A  bag  contains  6  red  balls,  5  white  balls,  and  4 
black  balls ;  find  the  probability  of  drawing : 

(a.)  One  red  ball. 

(6.)  Two  white  balls, 

(c.)  Four  red  balls. 

(cZ.)  One  ball  of  each  color, 

(e.)  Two  red  and  three  white  balls. 

(/)  One  red,  four  white,  and  two  black  balls. 

{g.)  Three  balls  of  each  color. 


396  COLLEGE   ALGEBRA. 

2.  If  out  of  every  1584  persons  living  at  tlie  age  of  14 
years,  1512  reacli  the  age  of  21,  what  is  the  probability  that 
a  person  aged  14  years  will  not  reach  the  age  of  21  ? 

3.  What  is  the  chance  of  throwing  doublets  in  a  single 
throw  with  two  dice  ? 

4.  What  is  the  chance  of  throwing  at  least  one  ace  in  a 
single  throw  with  tAvo  dice  ? 

5.  A  bag  contains  25  tickets  numbered  1,2,  3,  ...,  25;  if 
six  tickets  are  drawn,  find  the  probability : 

(a.)    That  4,  15,  and  21  will  be  among  them. 
(&.)    That  either  4,  15,  or  21  Avill  be  among  them. 

6.  If  four  cards  are  drawn  from  a  pack,  what  is  the 
probability  that  there  will  be  one  of  each  suit  ? 

7.  If  four  cards  .are  drawn  from  a  pack,  what  is  the  prob- 
ability that  they  will  be  the  ace,  king,  queen,  and  knave  of 
the  same  suit  ? 

8.  A  man  has  3  shares  in  a  lottery,  in  which  there  are 
3  prizes  and  7  blanks ;  find  his  chance  of  drawing  a  prize. 

554.  If  an  event  can  hap^Jen  in  ttco  or  more  independent 
loayjs  ivhose  respective  jprohahilities  are  known,  the  j^'^'ohahility 
of  the  haj)pening  of  the  event  is  equal  to  the  sum  ofthepiroha- 
hilities  of  its  hai^pening  in  the  separate  ways. 

Suppose  that  a  certain  event  can  happen  in  a  particular 
way  a  times  out  of  b,  and  fail  to  happen  in  this  particular 
way  b  —  a  times ;  suppose  also  that  the  same  event  can 
happen  in  another  way  a'  times  out  of  b,  and  fail  to  happen 
in  this  way  b  —  a'  times  ;  all  these  ways  being  equally  likely 
to  occur. 

Also,  suppose  that  the  different  ways  in  which  the  event 
can  happen  are  independent;  that  is,  if  the  event  happens  in 
tlie  first  way,  it  cannot  at  the  same  time  happen  in  the  second 
way. 


PROBABILITY.  397 

Then  the  event  happens  a  +  a'  times  out  of  b,  and  the 

probability  of  its  happening  is  (Art.  549),  or  -  +— • 

b  b       b 

T>ut  -  is  the  probabilitv  that  tlie  event  happens  in  the 

^  a'  ■ 

first  way,  and  —  is  the  probability  that  it  happens  in  the 

second  way. 

Hence,  the  probability  that  it  happens  is  equal  to  the  sum 
of  the  probabilities  of  its  happening  in  the  separate  ways. 

In  like  manner,  the  theorem  may  be  proved  to  hold  when 
there  are  more  than  two  independent  ways  in  which  the 
event  can  happen. 


555.  Exam2')le  1.  What  is  the  probability  of  throwing  4  in 
a  single  throw  with  a  pair  of  dice  ? 

The  event  can  happen  in  two  ways  ;  either  by  throwing  3 
and  1,  or  by  throwing  double  twos  ;  and  these  ways  are 
independent,  because  it  is  impossible  to  throw  3  and  1,  and 
double  twos  at  the  same  time. 

Each  die  can  come  up  in  6  ways ;  and  hence  the  pair  can 
be  thrown  in  6  X  6,  or  36  different  ways. 

3  and  1  can  be  thrown  in  two  ways,  for  the  first  die  may 
come  up  3  and  the  second  1,  or  the  first  die  may  come  up  1 
and  the  second  3 ;  hence  the  probability  of  throwing  3  and 

lisA. 

36 

Again,  double  twos  can  be  thrown  in  only  one  way  ;  hence 

the  probability  of  throwing  double  twos  is  — 

36 

..21  1 

Therefore  the  probability  of  throwing  4  is  •—  H ,  or  -— • 

36      36        12 

Example  2.  A  bag  contains  four  $  10  gold  pieces  and  six 
silver  dollars.  If  a  person  is  entitled  to  draw  two  coins  at 
random,  what  is  the  value  of  his  expectation  ? 


398  COLLEGE   ALGEBRA. 

Note.  If  a  person  has  a  chance  of  winning  a  certain  sum  of  money, 
the  product  of  the  sum  by  his  chance  of  winning  is  called  his  expecta- 
tion. 

The  number  of  combinations  of  the  four  gold  pieces,  taken 

2  at  a  time,  is  — ^,  and  the  number  of  combinations  of  the 

10-9 

ten  coins,  taken  2  at  a  time,  is  — - ;  hence  the  probability 

4-3  2 

of  drawinsr  two  erold  coins  is ,  or  — 

10-9'        15 

Then  the  value  of  the  expectation,  so  far  as  it  depends  on 

9  8 

the  drawing  of  two  gold  coins,  is  ^^  x  20,  or  -  dollars. 

15  3 

In  like  manner,  the  chance  of  drawing  two  silver  coins  is 

G-5  1 

-r^ — ,  or  - ;  and  the  value  of  the  corresponding  expectation 
10  •  9         3 

1  2 

is  -  X  2,  or  -  dollars. 
3  3 

Also,  the  chance  of  drawing  a  gold  coin  and  a  silver  coin 

is  (6  •  4)  -^ — ,  or  —^ ;  and  the  value  of  the  corresponding 

expectation  is  —  x  11,  or  —  dollars. 
lo  15 

Hence,  the  value  of  the  expectation  is  (  -  +  ^  +  —  J  dollars, 
or  $9.20.  ^^      ^      ^^^ 


EXAMPLES. 

556.    1 .  What  is  the  probability  of  throwing  6  in  a  single 
throw  with  two  dice  ? 

2.  What  is  the  probability  of  throwing  at  least  5  in  a 
single  throw  with  two  dice  ? 

3.  A  bag  contains  three  $  5  gold  pieces  and  five  silver 
dollars.     If  a  person  is  entitled  to  draw  one  coin  at  random, 

what  is  the  value  of  his  expectation  ? 


PROBABILITY.  399 

4.  A  lottery  has  34  prizes ;  four  of  $  500,  ten  of  $  250, 
and  twenty  of  f  25.  If  the  whole  number  of  tickets  is  100, 
what  is  the  value  of  each  ? 

5.  What  is  the  probability  of  throwing  15  in  a  single 
throw  with  three  dice  ? 

6.  What  is  the  probability  of  throwing  11  in  a  single 
throw  with  three  dice  ? 

7.  A  bag  contains  six  $5  gold  pieces,  and  four  other 
coins  which  have  all  the  same  value.  If  the  expectation  of 
drawing  two  coins  at  random  is  worth  $8.40,  what  is  the 
value  of  each  of  the  unknown  coins  ? 

8.  A  bag  contains  six  half-dollars,  six  quarter-dollars,  and 
six  dimes.  If  a  person  is  entitled  to  draw  three  coins  at 
random,  what  is  the  value  of  his  expectation  ? 


COMPOUND  EVENTS. 

557.  If  there  are  two  independent  events  ivJiose  respective 
prohahilities  are  known,  the  prohability  that  both  ivill  happen 
is  equal  to  the  product  of  their  separate  probabilities.     • 

Note.  Two  events  are  said  to  be  independent  when  the  occurrence 
of  one  is  not  affected  by  the  occurrence  of  the  other. 

Let  a  be  the  number  of  ways  in  which  the  first  event  can 
happen,  and  b  the  number  of  ways  in  which  it  can  fail ;  all 
these  ways  being  equally  likely  to  occur. 

Also,  let  a'  be  the  number  of  ways  in  which  the  second 
event  can  happen,  and  h'  the  number  of  ways  in  which  it 
can  fail ;  all  these  ways  being  equally  likely  to  occur. 

We  may  associate  together  any  one  of  the  a  +  b  cases  in 
which  the  first  event  happens  or  fails,  and  any  one  of  the 
a'  -\-  b'  cases  in  which  the  second  happens  or  fails ;  hence 
there  are  (a  -f  b)  (a'  -f  b')  cases,  equally  likely  to  occur. 

In  au'  of  these  cases  both  events  happen. 


400  COLLEGE   ALGEBRA. 

Therefore  the  probability  that  both  events  hapjoen  is 
aa' 


(a  +  h)  («'  +  &') 

But  IS  the  i^robability  that  the  first  event  happens, 

a  +  h 

and '■ — -  is  the  probability  that  the  second  happens. 

Hence,  the  probability  that  both  events  happen  is  equal 
to  the  product  of  their  separate  probabilities. 

And  in  general,  if  pi,  po,  Ps,  . . .,  are  the  respective  proba- 
bilities of  any  number  of  independent  events,  the  probability 
that  all  the  events  happen  is  Piihlh  •  •  •  • 

558.  Example  1.  Find  the  probability  of  throwing  an  aee 
in  the  first  only  of  two  successive  throws  with  a  single  die. 

The  probability  of  throwing  an  ace  at  the  first  trial  is  — 

5 

The  probability  of  not  throwing  one  at  the  second  trial  is  -• 

G 

Hence,  the  probability  of  throwing  an  ace  in  the  first 

only  of  two  successive  throws  is  -  x  -,  or  • — 
^  6      G'        3G 

Example  2.  Required  the  probability  of  throwing  an  ace 
at  least  once  in  three  throws  with  a  single  die. 

There  will  be  an  ace  unless  there  are  three  failures. 

5 
The  probability  of  failing  at  the  first  trial  is  - ;  and  this 

6 
is  also  the  probability  of  failing  at  each  of  the  other  trials. 

Hence  the  probability  that  there  will  be  three  failures  is 

5^5  5  125 
-  X  -  X  -,  or  — — 
G      G      G         216 

Then  the  probability  that  there  will  not  be  three  failures 

IS  1 ;  (Art.  550),  or  —  ^• 

21G  21G 


PROBABILITY.  401 

Example  3.  A  bag  contains  5  red  balls,  4  white  balls, 
and  3  black  balls.  Three  balls  are  drawn  in  succession, 
each  being  replaced  before  the  next  is  drawn.  What  is  the 
probability  that  the  balls  drawn  are  one  of  each  color  ? 


The  probability  that  the  first  ball  is  red  is  —  ;  the  proba- 

oi 
1 


bility  that  the  second  is  white  is  — ,  or  - ;  and  the  r)roba- 
^  12         3'  ^ 

bility  that  the  third  is  black  is    — ,  or     - 
•>  12'        4 

Hence  the  probability  of  drawing  a  red  ball,  a  white  ball, 

5       11  5 

and  a  black  ball,  in  this  assinned  order,  is  -—  x  -  X  -,  or  -- — 
■^  '      12      3      4        144 

But  a  red  ball,  a  white  ball,  and  a  black  ball  may  be 

drawn  in  [3,  or  G  different  orders  (Art.  543) ;  and  in  each 

5 
case  the  probability  is  — — 

Then  by  Art.  554,  the  probability  of  drawing  a  red  ball, 
a  white  ball,  and  a  black  ball,  without  regard  to  the  order 

5  5 

in  which  they  are  drawn,  is  x  G,  or  ■ — 

^  '       144        '        24 

559.  The  probability  of  the  concurrent  happ)ening  of  tivo 
dependent  events  is  equal,  to  the  probability  of  the  first,  multi- 
plied by  the  probability  that  ivhen  the  first  has  happened  the 
second  loill  folloiv. 

Let  a  and  b  have  the  same  meanings  as  in  Art.  557. 

Also,  suppose  that,  after  the  first  event  has  happened,  a' 
represents  the  number  of  ways  in  which  the  second  will 
follow,  and  b'  the  number  of  ways  in  Avhich  it  will  not  fol- 
low ;  all  these  ways  being  equally  likely  to  occur. 

Then  there  are  in  all  (o  +  b)  («'+  b')  cases,  ecpially  likely 
to  occur,  and  in  aa'  of  these  both  events  happen. 

Therefore  the  probability  that  both  events  happen  is 


(a  +  b){u'+b') 


402  COLLEGE    ALGEBRA. 

Hence  the  probability  that  both  events  happen  is  equal 
to  the  i^robability  of  the  first,  multiplied  by  the  probability 
that  when  it  has  happened  the  second  will  follow. 

And  in  general,  if  there  are  any  number  of  dependent 
events  such  that  2>i  is  the  probability  of  the  first,  2'>2  the 
probability  that  when  the  first  has  happened  the  second 
will  follow,  pg  the  probability  that  when  the  first  and  sec- 
ond have  hai)pened  the  third  will  follow,  and  so  on,  then  the 
probability  that  all  the  events  happen  is  j^iP-ilh  ■•  • 

560.  Example  1.  Let  it  be  required  to  solve  Ex.  3,  Art. 
558,  when  the  balls  are  not  replaced  after  being  drawn. 

The  probability  that  the  first  ball  is  red  is  -^ ;  the  prob- 
ability that  the  second  is  white  is  — ;   and  the  probability 

3 

that  the  third  is  black  is  — 
10 

Hence  the  probability  of  drawing  a  red  ball,  a  white  ball, 

and  a  black  ball,  in  this  assigned  order,  is  -^  x  —  X  — 

12      11      10 

But  the  balls  may  be  drawn  in  [3,  or  G  different  orders. 

Therefore  the  probability  of  drawing  a  rod  ball,  a  white 
ball,  and  a  black  ball,  without  regard  to  lli'.'  order  in  which 

they  are  drawn,  is  ^  x  —  X  —  X  u,  or  — 
^  12      11      10        '        11 

Example  2.  An  urn  contains  5  white  balls  and  3  black 
balls ;  another  urn  contains  4  white  balls  and  7  black  balls. 
AYhat  is  the  probability  of  obtaining  a  white  ball  by  a  single 
drawing  from  one  of  the  urns  taken  at  random  ? 

Since  the  urns  are  ('iiiially  lik(>ly  to  be  taken,  the  prol)a- 
bilily  ol'  t;il<iiig  tlic  lirst  iini  is  -;  and  tli(>  ])r()b;il)ility  of 
ilini   dniuin-  :i.  while  ImU  in,m  it  is  -• 


PROBABILITY.  403 

Hence  the  probability  of  obtaining  a  white  ball  from  the 

nrst  urn  is  -  X  -,  or  — • 
2      8         16 

In  like  manner,  the  probability  of  obtaining  a  white  ball 

14  2 

from  the  second  urn  is  -  x  — ,  or  —  • 
A       11  11 

5       2  87 

Hence  the  required  probability  is 1 ,  or  

16      11         176 


561.  Given  the  probability  of  the  hapjnnitig  of  an  event  in 
one  trial,  to  find  the  probability  of  its  happening  exactly  r  times 
in  n  trials. 

Let  p  be  the  probability  of  the  happening  of  the  event  in 
one  trial. 

Then  1  —  j)  is  the  probability  of  its  failing  (Art.  550). 

The  probability  that  the  event  will  happen  in  each  of  the 
first  r  trials,  and  fail  in  each  of  the  remaining  n  —  r  trials, 
is  jp'-(l-p)"-^. 

But  the  number  of  ways  in  which  the  event  may  happen 
exactly  r  times  in  n  trials  is  equal  to  the  number  of  com- 
binations of  n  things  taken  r  at  a  time,  or 

n{n-l):.{n-r  +  l)   ^^^^_  ^^^ 

Hence  the  probability  that  the  event  will  happen  exactly 
r  times  in  n  trials  is 

n(n-l)...(H-r  +  l)^.^^_^^„_.^  ^^^ 

[r 

For  example,  putting  r  =  1,  the  probability  that  the  event 
will  happen  exactly  once  in  n  trials  is  np{l  —pY~^ ;  putting 
r  =  2,  the  probability  that  the  event  will  happen  exactly 

twice  in  n  trials  is  ^^^^~    '  P^O-  —pY~^'i  and  so  on. 


404  COLLEGE   ALGEBRA. 

In  like  manner,  tlie  probability  that  tlie  event  will  fail 
exactly  r  times  in  n  trials  is 

\l 

562.  Given  the  probability  of  tlie  happening  of  an  event  in 
one  trial,  to  find  the  probability  of  its  hapjyening  at  least  r  times 
in  n  trials. 

The  event  happens  at  least  r  times  if  it  happens  exactly 
n  times,  or  fails  exactly  once,  twice,  . . .,  n  —  r  times. 

Therefore  the  probability  that  it  happens  at  least  r  times 
is  equal  to  the  sum  of  the  probabilities  of  its  happening 
exactly  n  times,  or  failing  exactly  once,  twice,  . . .,  n  —  r 
times ;  which,  by  Art.  561,  is 

2r+npf\l  -p)+...  -f-^A ^ '^    ^    -'jj'(l-j))"  ••. 

\n  —  r 

563.  ExampAe.  A  bag  contains  five  tickets  numbered  1, 
2,  3,  4,  5.  Five  tickets  are  drawn  at  random,  each  being 
replaced  before  the  next  is  drawn.  Find  the  probability  of 
drawing  the  ticket  marked  1  exactly  three  times,  and  at  least 
three  times. 

In  this  case,  p  =  --,  r  =  3,  n  =5. 
5 

Then  by  Art.  561,  (1),  the  probability  of  drawing  the 

ticket  marked  1  exactly  three  times  is 

l-2-3\5j\5j'        625 

And  by  Art.  562,  the  probability  of  drawing  it  at  least 
three  times  is 


(sj^^m 


l-'Ar,   {u   •       313S 


PROBABILITY.  495 


EXAMPLES. 


564.  1.  If  eight  coins  are  tossed  up,  what  is  the  chance 
that  one  and  onjy  one  will  turn  up  head  ? 

2.  A  purse  contains  one  dollar  and  three  dimes ;  another 
contains  two  dollars  and  four  dimes ;  a  third  three  dollars  and 
one  dime.  What  is  the  chance  of  obtaining  a  dollar  by  draw- 
ing a  single  coin  from  one  of  the  purses  taken  at  random  ? 

3.  What  is  the  probability  of  throwing  exactly  three 
aces  in  five  throws  with  a  single  die  ? 

4.  What  is  the  probability  of  throwing  at  least  three 
aces  in  five  throws  with  a  single  die  ? 

5.  If  three  cards  are  drawn  from  a  pack,  what  is  the 
chance  that  they  will  consist  of  a  king,  queen,  and  knave  ? 

6.  The  probability  that  A  can  solve  a  certain  problem  is 
f,  and  the  probability  that  B  can  solve  it  is  f.  Find  the 
probability  that  the  problem  will  be  solved  if  both  try. 

7.  If  a  coin  is  tossed  up  ten  times,  what  is  the  chance 
that  the  head  will  present  itself  exactly  five  times  ? 

8.  A  bag  contains  ten  tickets  numbered  0, 1,  2,  ...,  9.  If 
three  tickets  are  drawn  at  random,  what  is  the  probability 
that  their  sum  is  22  ? 

9.  Two  bags  contain  each  4  black  and  3  white  balls.  A 
ball  is  drawn  at  raiidom  from  the  first,  and  if  it  is  white,  it 
is  put  into  the  second  bag,  and  a  ball  drawn  at  random  from 
that  bag.     Find  the  chance  of  drawing  two  white  balls. 

10.  If  the  odds  are  5  to  3  against  a  person  who  is  now 
40  living  till  he  is  65,  and  11  to  6  against  a  person  who  is 
now  45  living  till  he  is  70,  what  is  the  chance  that  at  least 
one  of  these  persons  will  be  alive  25  years  hence  ? 

11.  What  is  the  probability  of  throwing  aces  with  a  paii 
of  dice  at  least  three  times  in  four  trials  ? 


406  COLLEGE    ALGEBRA. 

12.  A  bag  contains  5  v/liite  and  8  black  balls.  Two 
drawings,  each  of  3  balls,  are  made,  tlie  balls  first  drawn  not 
being  replaced  before  the  second  trial ;  what  is  the  chance 
that  the  first  drawing  will  give  3  white,  and  the  second  3 
black  balls  ? 

13.  A  bag  contains  ten  tickets,  five  numbered  1,  2,  3,  4,  5, 
and  the  rest  blank.  Three  tickets  are  drawn  at  random, 
each  being  replaced  before  the  next  is  drawn  ;  what  is  the 
probability  that  their  sum  is  10  ? 

14.  A's  skill  at  a  game  is  two-thirds  of  B's.  What  is 
the  chance  that  A  wins  at  least  two  games  out  of  five  ? 

16.  A  bag  contains  4  red,  3  white,  and  2  black  balls.  A 
ball  is  drawn  and  not  replaced.  Another  ball  is  then  drawn. 
Find  the  chance  that  the  two  balls  are  of  the  same  color. 

16.  A's  skill  at  a  game  is  double  B's.  What  is  the  prob- 
ability that  A  wins  four  games  before  B  wins  two  ? 

17.  A  bag  contains  6  red  balls,  5  white  balls,  and  4  black 
Ijalls.  Four  balls  are  drawn  in  succession,  and  are  not 
replaced  after  being  drawn.  What  is  the  chance  that  tAvo 
of  them  are  red,  one  white,  and  one  black  ? 

18.  If  one  vessel  out  of  every  ten  is  wrecked,  what  is 
the  chance  that,  out  of  five  vessels  expected,  at  least  four 
will  arrive  safely  ? 

19.  A  and  B  draw  in  succession,  in  the  order  named, 
from  a  purse  containing  three  sovereigns  and  four  shillings. 
Find  their  respective  chances  of  first  drawing  a  sovereign, 
the  coins  when  drawn  not  being  replaced. 

20.  A,  B,  and  C  draw  m  succession,  in  the  order  named, 
from  a  bag  containing  three  white  balls  and  five  black  balls. 
Find  their  respective  chances  of  first  drawing  a  white  ball, 
tlie  balls  Avhen  drawn  not  being  replaced. 


CONTINUED   FRACTIONS.  407 


XXXVIII.    CONTINUED  FRACTIONS. 

565.   A  continued  fraction  is  an  expression  of  the  form 
h 


a  + 


,     •  d 


e+  .-. 

or,  as  it  is  usually  written  in  practice, 
,     6      d 
c+e  +  -" 
We  shall  limit  ourselves  in  the  present  work  to  continued 
fractions  of  the  form 

,    1       1 

a  +  - ; 

6+C  +  .-- 
where  each  numerator  is  unity,  a  any  positive  integer  or  0, 
and  each  of  the  quantities  h,  c,  ...,  a  positive  integer. 

566.  A  terminating  continued  fraction  is  one  in  which 
the  number  of  denominators  is  finite ;  as, 

a-{ 

b+  c+d 

It  may  be  reduced  to  an  ordinary  fraction  by  the  process 
of  Art.  1G9. 

An  infinite  continued  fraction  is  one  in  which  the  number 
of  denominators  is  indefinitely  great. 

567.  In  the  continued  fraction 

,11         1 


a  +ao+a,+ 


Qi  is  called  the  Jirst  convergent ; 

ai  -) is  called  the  second  convergent; 

ttj  H is  called  the  third  convergent ;  and  so  on. 

a.,+  ((■■ 


408  COLLt:GE   ALGEBRA. 

Note.    If  «!  =  0,  as  in  the  continued  fraction 
J ^       1 

then  0  is  considered  tlie  first  convergent. 

568.  Ayiy  ordinary  fraction  in  its  loivest  terms  may  he  coih- 
uerted  into  a  terminating  continued  fraction. 

Let  the  given  fraction  be  -,  where  a  and  6  are  prime  to 
each  other. 

Divide  a  by  h,  and  let  a^  denote  the  quotient  and  hi  the 
remainder ;  then, 

a  ,  &,  ,1 

-  =  0,+  — =  a,  H — 
h  h         '      6_ 

Divide  h  by  6i,  and  let  a<i  denote  the  quotient  and  62  the 
remainder ;  then, 

a  ,        1  ,1 

^  «.  +  ^  a,  +  A 

Again,  divide  h^  by  &2)  ^-'id  1^^  f<3  denote  the  quotient  and 
63  the  remainder  ;  then, 

a  ,  1  ^1 

6^^'-+       ,        1      =^'+       ,       1      • 


«2  H r  "^  + 


"3  +  r  "3+- 

02  02 

&3 

The  process  is  the  same  as  that  of  finding  the  Highest 
Common  Factor  of  a  and  h  (Art.  139) ;  and  since  a  and  h 
are  prime  to  each  other,  we  must  eventually  obtain  a  remain- 
der unity,  at  which  point  the  o})eration  terminates. 

Hence  any  ordinary  fraction  in  its  lowest  terms  can  be 
converted  into  a  terminating  continued  fraction. 


CONTINUED   FRACTIONS.  409 

62 
Example.   Convert  ^  into  a  continued  fraction. 

23)62(2  =  ai 
46 

16)23(1  =  a2 
16 
7)16(2  =  a3 
14 
■  2)7(3  =  a4 

1 

Therefore,  ^=2  +  ^  — .l.L 

'  23  1  +  2+3+2 


569.   A  quadratic  surd  (Art.  291)  may  be  converted  into 
an  infinite  continued  fraction. 

Example.    Convert  V6  into  a  continued  fraction. 

The  greatest  integer  in  V6  is  2 ;  we  then  write 

V6  =  2  +  (V6-2). 

Reducing  V6  —  2  to  an  equivalent  fraction  with  a  rational 
numerator  (Art.  308),  we  have 

V6  =  2  +  (-^-^)(^^  +  ^)=2+       - 

V6  +  2  V6  +  2 

=  2+—^ (1) 

V6  +  2 


The  greatest  integer  in  — — ^t_  is  2 ;  we  then  write 
V6  +  2^^  I  V6-2 


2 


_^  ,   (V6-2)(V6+2)^^  , 


2(V6  +  2)  V6+2 


410  COLLEGE   ALGEBRA. 

Substituting  in  (1), 

V6  =  2  + ^ (2) 

-v'6  +  2 

The  greatest  integer  in  VC  +  2  is  4 ;  we  then  write 

V6  +  2  =  4+(V(3-2)^4  +  (^--)(^''  +  ^) 

V6  +  2 


V6  +  2  VC  +  2 

2 
Substituting  in  (2),  we  have 

1 


V6  =  2  + 


2  + 


4  + 


V6  +  2 


2 
The  steps  now  recur,  and  we  have 

V6=2+l     1     1       1 


2+4+2+4+. 


Note.   An  infinite  continued  fraction  in  which  the  elements  recur 
is  called  a  periodic  continued  fraction. 

570.  A  x>eriodic  continued  fraction  may  alivays  be  expressed 
as  the  root  of  a  certain  quadratic  equation. 

Example.    Express — as  the  root  of  a  certain 

^  ^         1  +  3  +  1  +  3+.. . 

quadratic  equation. 

Let  X  denote  the  vahie  of  the  fraction ;  then, 

=  J_     1      ^     S  +  x     ^  S  +  x 
1  +  o  +  x      3  +  x  +  l      4  +  X 

Clearing  of  fractions, 

4:X  -]-  X-  =  3  +  X,  or  X-  +  ."5  x  =  3. 


CONTINUED   FRACTIONS.  411 

Solving  the  equation, 

2  2 

Note.    The  +  sign  is  taken  before  the  radical,  since  x  must  evi- 
dently be  a  positive  quantity. 


PROPERTIES   OF   CONVERGENTS. 

571.    Let  the  continued  fraction  be 
111  1 


+ 


and  let  p^  denote  the  numerator,  and  q^  the  denominator,  of 
the  rth  convergent  (Art.  567)  when  expressed  in  its  sim- 
plest form. 

572.  To  determine  the  law  of  formation  of  the  successive 
convergents. 

The  first  convergent  is  aj. 

The  second  is  cti  -| =    ^  '-        • 

a.,  a.2 

rri     4-1  •    1  •             ,11            ,        ^'".  a,aoa.,-^-ai-\-ao 

The  third  is     Ui  -\ =  f'l  H —  =    ^  '  '^  — — — ^• 

tto+ag  a/<3+l  ttoGg+l 

The  third  convergent  may  be  written  in  the  form 

(aia2  +  1)0'^  + cii . 
a^ttg  + 1         ' 

in  which  we  observe  that : 

1.  The  numerator  is  equal  to  the  numerator  of  the  preced- 
ing convergent,  mnltipUed  by  the  last  denominator  taken,  plus 
the  numerator  of  the  convergent  next  but  one  preceding. . 

2.  Tlie  denominator  is  equcd  to  the  denominator  of  the  pre- 
ceding convergent,  multiplied  by  the  last  denominator  taken, 
plus  the  denominator  of  the  convergent  next  but  one  preceding. 

We  will  now  prove  by  Induction  (Note,  p.  46)  that  the 
above  laws  hold  for  all  convergents  after  the  second,  when 
expressed  in  their  simplest  forms. 


412  COLLEGE   ALGEBRA. 

Assume  that  the  laws  hold  for  all  convergents  as  far  as 
the  ?ith. 

The  nth  convergent  is 

&=„.+^^...i. 

q^  a2+a3+      a„ 

Then  since  the  last  denominator  is  a„,  we  have 

Pn  =  anPn-l+Pn~2,     and    qn  =  Ct^Qn^l  +  Qn-H-  (1) 

Whence,  Pn  ^CinPn-^+Pn-,^  (2) 

q.       Cl„q„__,+  q,_2 

The  {n  4-l)st  convergent  is 

J_  J.  1      1 

which   differs  from  the  ?ith  only  in  having  a„  -\ — ,   or 

^'''^"+1  +  ^,  in  place  of  a,. 

Substituting  ^^"^^"+1  +  ^  for  a„  in  (2),  we  have 


l+i>n-2 

?«+l        «,/<„+!  +  1  ^^ 

1  +  g»-2 

a»+i(««9«-i  +  <?«  2)  +  g„-i 

^  «,.+lPn  +  />n-l^    by    (1).  (3) 

It  is  evident  that  the  second  member  of  (3)  is  the  sim- 
plest form  of  the  (n  +  l)st  convergent,  and  therefore 

Pn+i  =  an+iPn+p„-i,  aud  qn+i  =  a,ni(]„  +  (In-i- 
These  results  are  in  accordance  witli  the  laws  stated  on 

the  preceding  page. 

Hence,  if  the  laws  hold  for  all  convergents  as  far  as  the 

7ith,  they  also  hold  as  far  as  the  (n  +  l)st. 


CONTINUED   FRACTIONS.  413 

But  we  know  that  they  hold  as  far  as  the  third  convergent, 
and  hence  they  also  hold  as  far  as  the  fourth  ;  and  since  they 
hold  as  far  as  the  fourth,  they  also  hold  as  far  as  the  fifth ; 
and  so  on. 

Therefore  the  laws  hold  for  all  convergents  after  the 
second. 

Examx>le.    Find  the  first  five  convergents  of 

1  +  2  +  3+4+.. . 

The  first  convergent  is  1,  and  the  second  is  1  + 1,  or  2. 

Then  by  aid  of  the  laws  just  proved, 

^1     ^T   1  ■      2.2  +  1        5 
the  third  is  —     • 


the  fourth  is 
the  fifth  is 


1. 

2  +  1 

3' 

5. 

•  3  +  2 

17 

3. 

.3  +  1 

10' 

17 . 4  +  5 

73 

10-4  +  3      43 


573.  The  difference  between  two  consecutive  convergents 
£1'  and  -L^  is  equal  to •. 

g«        qn+i  Quqn+i 

The  difference  between  the  first  and  second  convergents  is 

«i  +  -    -  tti  =  -. 
a.,/  c(2 

Thus  the  theorem  holds  for  the  first  and  second  conver- 
gents. 

Assume  that  it  holds  for  the  nth.  and  (n  +  l)st  convergents  ; 
that  is, 

f^ ~f^'  =  ^,  or  p„q,^_,  ~,.„,,g,.  =  1.  (1) 

Qn  y-.  +  l  UnQn+l 

Then, 


414  COLLEGE   ALGEBRA. 

yn+iyn+2  yn+l5H+2 

Hence  if  the  theorem  holds  for  any  pair  of  consecutive 
convergents,  it  also  holds  for  the  next  pair. 

But  we  know  that  it  holds  for  the  first  and  second  conver- 
gents, and  hence  it  also  holds  for  the  second  and  third ;  and 
since  it  holds  for  the  second  and  third,  it  also  holds  for  the 
third  and  fourth ;  and  so  on. 

Therefore  the  theorem  holds  universally. 

574.  It  follows  from  Art.  573  that  j9„  and  g„  can  have  no 
common  divisor  except  unity ;  for  if  they  had,  it  would  be  a 
divisor  of  p„  g„+i  ~  p„+i  g„,  or  unity,  which  is  impossible. 

Therefore  all  convergents  formed  in  accordance  with  the 
laws  of  Art.  572  are  in  their  lowest  terms. 

575.  The  even  convergents  are  greater,  and  the  odd  con- 
vergents less,  than  the  fraction  itself. 

I.    The  first  convergent,  a^,  is  less  than  the  fraction  itself, 
1 


is  omitted. 


ao+' 


II.  The  second,  a^  -{ ,  is  greater,  because  its  denomina- 
tor cia  is  less  than  a,  -\ ,  the  denominator  of  the  fraction. 

III.  The  third,  a.  H ,  is  less,  because,  by  II.,  the 

I        ch+a-I  11 

denominator   a.,  -\ is    greater   than   a,  -\ ,    the 

Ug         ^  a^+a^-i 

denominator  of  the  fraction  ;  and  so  on. 

Hence  the  first,  third,  ...,  convergents  are  less,  and  the 
second,  fourth,  ...,  convergents  greater  than  the  fraction 
itself. 


CONTINUED    FRACTIONS. 


415 


576.   Any  convergent  is  nearer  than  the  2'>rececUng  convergent 
to  the  value  of  the  fraction  itself. 

By  Art.   572,  Pn+2^<^n+2Pn+l+ 2^.^ 

qn+2         an+oQu+l-hgn 

The   fraction   itself  is  obtained  from   its    (?i-|-2)d  con- 
1 


vergent  by  putting  a„+2 


in  place  of  a„+2- 


«n+3  + 

Hence,  denoting  the  vahie  of  the  fraction  itself  by  a;,  we 


have 


i>„4-l  +Pn 

q»+i  +  ?« 

mPn+\-\-Pn 


where  m  stands  for  a,.+2  + 


ISTow, 


tMu+i  +  qn    qn 

^{Pn+iqn-Pnqn+l) 


qn{mq„^,  +  qn) 


qn{^Mn+l  +  qn) 


(Art.  573). 


(1) 


Also,  X 


Since 


Pn+l^'>nPn+l+Pn  _Pn+\ 

«ig,-+i  +  qn  ~  qn+i 

Pnqn+l-^Pn+iqn 


qn+i 


qn+1  (mg„+i  +  g„)      g„+i  (mg„+i  +  q„ ) 
1 


is   a  positive  integer,  a,^_^.2  + 


CW3+' 


(2) 
is>l; 


that  is,  m  is  >  1. 

And  since  q„^i  =  a„_^_lq„  +  q„_]^  (Art.  572),  g„^i  is  >  q„. 

Therefore  the  fraction  (2)  is  less  than  the  fraction  (1), 
for  it  has  a  smaller  numerator  and  a  greater  denominator. 

Hence  the  (n  +  l)st  convergent  is  nearer  than  the  ?ith  to 
the  value  of  the  fraction  itself. 


416  COLLEGE    ALGEBRA. 

577.    By  Art.  576,  the  difference   between   the  fraction 
itself  and  its  nth.  convergent  is 

m  1 


gn(m^n+i  +  g,.) 


9«(^^+i  +  | 


(1) 


Since  m  is  >  1  (Art.  576),  the  denominator  5„f  g„+i  +  ^] 

is    <qn(qn+l+gr,)- 

The  denominator  is  also  >q„q„+i- 

Hence  the  fraction  (1)  is  > -,  and  < 


qniqu+i  +  qn)-  qnqn+i 

That  is,  the  erroj-  made  in  taking  the  nth.  convergent  for 
the  fraction  itself  lies  between  the  limits 

1  and      1 


qn{qn+i  +  qn)       q.qn+1 


EXAMPLES. 


578.   Convert  each  of  the  following  into  a  continued  frac- 
tion, and  find  in  each  case  the  first  five  convergents  : 


1.  ^. 

39 

3.  3.61. 

5.  J^. 

326 

7  436 
■  345' 

2.  ^. 
91 

4.  11-2. 
153 

6.  Ml. 

89 

g  3015 
6961 

Convert  each  of  the  follovv^ing  into  a  continued  fraction, 
find  in  each  case  the  first  four  convergents,  and  determine 
limits  to  the  error  made  in  taking  the  third  convergent 
for  the  fraction  itself  : 

9.    V5.  12.    V3.  15.    2V5.  18.    Vi4. 

10.  ViO.         13.    VfJ-  16.    V7.  19.    V33. 

11.  V17.         14.    yj^.  17.    ^'^^  +  ^-    20.    2V35. 


CONTINUED   FRACTIONS.  417 

Express  each  of  the  following  in  the  form  of  a  surd : 


21. 

1111 

2  +  0+2  +  0+--- 

23.    2  +  ^-^—- 
1+1  +  ... 

22. 

1111 

1  +  4+1  +  4+-" 

24.    3+111       1     ^ 

1  +  8+1  +  8+.. . 

25.  The  ratio  of  the  circumference  of  a  circle  to  its  diame- 
ter is  approximately  equal  to  3.14159;  express  this  decimal 
as  a  continued  fraction,  and  find  the  first  four  convergents. 

26.  The  modulus  of  the  common  system  of  logarithms  is 
approximately  equal  to  .43429;  express  this  decimal  as  a 
continued  fraction,  find  its  seventh  convergent,  and  deter- 
mine limits  to  the  error  made  in  taking  this  convergent  for 
the  fraction  itself. 

27/  The  base  of  the  Napierian  system  of  logarithms  is 
2.7183  approximately ;  express  this  decimal  as  a  continued 
fraction,  find  its  eighth  convergent,  and  determine  limits  to 
the  error  made  in  taking  this  convergent  for  the  fraction 
itself. 

28.  Express  the  positive  root  of  the  equation 

^.2  _  3,  _  ;L]^  ^  0 

as  a  continued  fraction,  and  find  the  first  five  convergents. 

Convert  each  of  the  following  into  a  continued  fraction, 
and  find  in  each  case  the  first  four  convergents : 

29.  Vl3.         30.    -^.         31.    V9cr  +  3.         32.    V22. 

V33 

33.    Exi)ress  «  -| in  the  form  of  a  surd. 


418  COLLEGE   ALGEBRA. 


XXXIX.    SUMMATION  OF  SERIES. 

579.  The  Summation  of  an  infinite  literal  series  is  the 
process  of  finding  a  finite  expression  from  which  the  series 
may  be  developed. 

The  result  represents  the  series  only  for  such  values  of 
the  letters  involved  as  make  the  series  convergent. 

A  method  has  already  been  given  (Art.  429)  for  finding 
the  sum  of  an  infinite  geometrical  series. 

RECURRING  SERIES. 

580.  A  Recurriiig  Series  is  an  infinite  series  of  the  form 

a-o  +  a^x  +  agO^  +  •  •  •? 
where  any  r  +  1  consecutive  terms  are  connected  by  a  rela- 
tion of  the  form 

p,  q,  ,..,  s  being  constants. 

The  above  recurring  series  is  said  to  be  of  the  rth  order, 
and  the  expression 

1  4-^.13  +  gx^  +■••  +  sx" 

is  called  its  scale  of  relation. 

For  example,  in  the  infinite  series 

any  three  consecutive  terms  are  so  related  that  the  last 
term,  plus  —2x  times  the  preceding  term,  plus  x-  times  the 
next  but  one  preceding,  is  equal  to  0. 

Hence  the  series  is  a  recurring  series  of  the  second  order, 
and  its  scale  of  relation  is  1  —2x  -{-a^. 

Note.  An  infinite  geometrical  series  is  a  recurring  series  of  the 
first  order. 


SUMMATION   OF   SERIES.  '  419 

581.  To  find  the  scale  of  relation  of  a  recurring  series. 

If  the  series  is  of  the  first  order,  the  scale  of  relation 
may  be  found  by  dividing  any  term  by  the  preceding  term, 
and  subtracting  the  result  from  1. 

If  the  series  is  of  the  second  order,  and  ao,  ctj,  a2,  a^,  ..., 
are  its  consecutive  coefficients,  and  1  -\-px  +  qa?  its  scale  of 
relation,  we  shall  have 

1  a.  +pa2  -t-  gcti  =  0  ; 
from  which  p  and  q  may  be  determined. 

If  the  series  is  of  the  third  order,  and  ao,  aj,  a^,  a-s,  ^4, 
tts,  ...,  are  its  consecutive  coefficients,  and  1  +  ^j.t  +  g;z;^  +  ?-ar' 
its  scale  of  relation,  we  shall  have 

r  ttg  +pa2  +  qai  +  ra^  =  0, 
\  a^  +pa^  +  qa^  -f  ra^  =  0, 
[  «5  +  2^«4  +  qa.^  +  ra.^  =  0  ; 
from  which  p,  q,  and  r  may  be  determined. 

It  is  evident  from  the  above  that  the  scale  of  relation  of 
a  recurring  series  of  the  rth  order  may  be  determined  when 
any  2r  consecutive  terms  are  given. 

To  ascertain  the  order  of  a  series,  we  may  first  make  trial 
of  a  scale  of  relation  of  three  terms  ;  if  the  result  does  not 
correspond  with  the  series,  try  a  scale  of  four  terms,  five 
terms,  and  so  on  until  the  true  scale  of  relation  is  found. 

If  .the  series  is  assumed  to  be  of  too  high  an  order,  the 
equations  corresponding  to  the  assumed  scale  will  not  be 
independent.     (Compare  Art.  215.) 

582.  To  find  the  sum  of  a  recurring  series  iclien  its  scale  of 
relation  is  Tcnoion. 

Let  l-\-px  +  qx?  be  the  scale  of  relation  of  the  series 


420  COLLEGE   ALGEBRA. 

Denoting  the  sum  of  the  first  n  terms  by  S,„  we  have 

>S„  =  Oo  +  «i x-\-  a.^x-  -\-  ■••  +  a„_ix"  ^ 
Whence, 

and        qx^Sn  =  gaoX^  +  •••  +ga„_3ic''^^  +  qa,^^2^''  +  5a„_ia;"+\ 

Adding  these  equations,  and  remembering  that,  by  virtue 
of  the  scale  of  relation, 

«2  -\-p(h  +  Q^O  =0,    •••,  a„_i  -}-J9a„_2  +  (?«n-3  =  0, 

we  have 

=  tto  +  («!  +i:'«o)a:  -f  (pa„-i  +  5«»-2)^"  +  5«n-ia;"+^ 
Whence, 

which  is  a  formula  for  the  sum  of  the  first  n  terms  of  a 
recurring  Series  of  the  second  order. 

If  X  is  so  taken  that  the  given  series  is  convergent,  the 
expression 

(i5o„_i  +  qcin-2)^'^  +  ga„_ia;''+^ 

approaches  the  limit  0  when  n  is  indefinitely  increased,  and 
(1)  becomes 

l+px+qaf    '  ^^^ 

which  is  a  formula  for  the  sum  (Art.  579)  of  a  recurring 
series  of  the  second  order. 

li  q  =  0,  the  series  is  of  the  first  order,  and  therefore 
«i  +iJao  =  0  ;  whence, 

s  =  -^^ ;  (3) 

I+JIX 

which  is  a  formula  for  the  sum  of  a  recurring  series  of  the 
first  order.      (Compare  Art.  129.) 


SUMMATION    OF   SERIES.  421 

In  like  manner,  we  shall  find  the  formula 

^  ^  a,j  +  (cti  +pao)x  +  (cio  +P%  +  gfto)ar^  r^\ 

1  +px  +  ga;^  +  rar' 

for  the  sum  of  a  recurring  series  of  the  third  order. 

Note.  It  will  be  observed  in  each  case  that  the  denominator  of  the 
fraction  is  the  scale  of  relation. 

583.  A  recurring  series  is  formed  by  the  expansion  in  an 
infinite  series  of  a  fraction,  called  the  generating  fraction. 

The  operation  of  summation  reproduces  the  fraction;  the 
process  being  the  reverse  of  that  of  Art.  471. 

584.  Example.     Find  the  sum  of  the  series 

1  +  Ox  -  15ar^  +  57x3  -  ISOa;"  +  •••  . 

To  determine  the  scale  of  relation,  we  first  assume  the 
series  to  be  of  the  second  order  (Art.  581). 

Substituting  a^  =  1,  a^  —  9,  a.^  =  —  15,  and  a^  =  57  in  the 
equations  of  Art.  581,  we  have 

■15+    9p+    g  =  0, 
57-15j9  +  9r^  =  0. 
Solving,  we  find  p  =  2  and  g  =  —  3. 

To  ascertain  if  1  +  2  x  —  3  a;-  is  the  true  scale  of  relation, 
consider  the  fifth  term. 

Since  -159x*+(2a;)  (57ar'^)  +  (-3x2)  (-15^2)  is  equal  to 
0,  it  follows  that  l  +  2x  —  3a:^is  the  true  scale. 

Substituting  the  values  of  a^,  a^  p,  and  q  in  (2),  Art.  582, 

^^l+(9  +  2)a;^      l  +  llx 
l+2a;-3ar^      l  +  2a;-3a;2' 

585.  It  is  possible,  by  aid  of  Art.  474,  to  find  an  expres- 
sion for  the  rth  term  of  the  series  of  Art.  584. 

.  1  +  llx  A      ,       B 

Assume,        ~ = 

l  +  2a;  — 3a;-      1-x      1+ox 


422  COLLEGE   ALGEBRA. 

Then,  1 +  llx  =  A(l  +  3x)  + B (1  -  x). 

Putting  a;  =1,  12  =  4, A;  whence  ^1  =  3. 

Putting  x  =  —  -,      — ^  =  -iJ;  whence  B  =  —  2. 
o  o       o 

Then,  1  +  11^     _     3  2 


l+2a;-3a"'      1  -  a;      l+Sic 

=  3{l-j-x  +  x^  +  x"+.:) 

-2ll+{-3x)  +  (-3xy+...]. 
Therefore  the  Hh  term  of  the  given  series  is 

3x^-1  -  2  (-  3xy-\  or  [3  -  2  (-  3)'-']x'-\ 

EXAMPLES. 

586.   In  each  of  the  following,  find  the"  generating  frac- 
tion, and  the  expression  for  the  Hh  term : 

1.  l  +  5x  +  19x'  +  65x'+211x*-\--'. 

2.  2-x-|-5.^--7.r'  +  17a;^ 

3.  l-4.x-2x'-10o^-Ux* 

4.  2  -  5x +^  17x^-650^  + 257 x' 

6.  3  +  5x-5a^-115a.-3_845.'c^ 

6.  5  +  8x  +  56x^-\-176x'-i-800x'+-" 

7.  i-x  +  61x'-319x'  +  2U9x* 

8.  l-13a;  +  89a^-517a^  +  2801a;* 

In  each  of  the  following,  find  the  generating  fraction,  and 
continue  the  series  to  two  more  terms : 

9.  i+3a;_ar_5ar^-7a;*-ar'  +  lla;''  + 

10.  1+ix-  7af  -  2x'  +  9.x-'  +  lO.x''  -  51  a;"  +  •••  • 

11.  2  -  7.«  +  iDx-  -  54  .*.-'  +  UGa;^  -  397.c-^'+  \i%7x''^ 


SUMMATION   OF   SERIES.  428 

THE  DIFFERENTIAL  METHOD. 

587.  If  the  first  term  of  any  series  is  subtracted  from 
the  second,  the  second  from  the  third,  and  so  on,  a  series  is 
formed  which  is  called  the  first  order  of  differences  of  the 
given  series. 

The  first  order  of  differences  of  this  new  series  is  called 
the  second  order  of  differences  of  the  given  series  ;  and  so  on. 

Thus,  in  the  series 

1,     8,«   27,     64,     125,     216,     ..., 
the  successive  orders  of  differences  are  as  follows : 

1st  order,        7,     19,     37,     61,     91,     .... 

2d  order,  12,     18,     24,     30,     .... 

3d  order,  6,       6,       6,       .... 

4th  order,  0,       0, 

588.  The  Differential  Method  is  a  method  for  finding  any 
term,  or  the  sum  of  any  number  of  terms  of  a  series,  by 
means  of  its  successive  orders  of  differences. 

589.  To  find  any  term  of  the  series 

tti,     tta,     ttg,     a^,     ...,     a„,     ftn+i}     •••  • 
The  successive  orders  of  differences  are  as  follows  : 
1st  order,     a^  —  a^,  a^  —  a.^,  a^  —  ar^,  •••,  a„+i  —  «„,  •••  • 
2d  orderj     ag  —  2a2  +  ai,  a4  —  2a3  +  a,,  •••  • 
3d  order,     a^  —  3 a^ -\- ^ a.^  —  a^,  •••;  etc. 
Denoting  the  first  terms  of  the  1st,  2d,  3d,  ...,  orders  of 
differences  by  dj,  d.,  d^,  ...,  respectively,  we  have 
di  =  a2  —  cti',  whence,  ctj  =  aj  +  c^i- 
^2  =  ag  —  2  ao  +  «! ;  whence, 

a^  =  -  a,+  2ao+  d.  =  -  a^+  2ai+  2di+  d.  =  «!+  2d,+  d^ 
cZg  =  a4  —  3  ag  +  3  tta  —  tti ;  whence, 
a^  =  cii  -  3a2  +  Sag  +  dg  t=  a,  +  3cZi  +  ScZa  +  dg ;  etc. 


424  COLLEGE   ALGEBRA. 

It  will  be  observed,  in  tlie  values  of  ag,  a^,  and  Ui,  that 
the  numerical  coefficients  of  the  terms  are  the  same  as  the 
coefficients  of  the  terms  in  the  expansion  by  the  Binomial 
Theorem  of  a  +  x  to  the  Jirsty  second,  and  third  powers, 
respectively. 

We  will  now  prove  by  induction  that  this  law  holds  for 
any  term  of  the  given  series. 

Assume  the  law  to  hold  for  the  nth  term,  a„ ;  then  the 
coefficients  of  the  terms  will  be  the  same  as  the  coefficients 
of  the  terms  in  the  expansion  by  the  IJinomial  Theorem  of 
a  +  X  to  the  (n  —  l)st  power. 

That  is, 

a„  =  a,  +  (n-l)cZ,+  (^^-^)(^^-^>d, 

\1 

\1 
If  the  law  holds  for  the  nth  term  of  the  given  series,  it 
must  also  hold  for  the  nth.  term  of  the  first  order  of  differ- 
ences ;  whence, 

a„^i-a„  =  d,  +  (n-l)d,+  ^''~^|^''~^^d3+-'     (2) 
Adding  (Ij  and  (2),  we  have 

a„+i  =  ai  +  [(n-l)  +  l]di  +  ^[(n-2)  +  2]d, 
^(n-l)(.-2)^^^_3^^3^^^^_^^,, 

,       ,    ,  n(n  —  l\  ,    ,  7i(n  —  l)(n  —  2)  ,    ,  ,o\ 

=  ai  +  ndi  +    V  ^    ^d.,-\-^ ^^ ^^3-1 (3) 

This  result  is  in  accordance  with  the  above  law. 

Hence,  if  the  law  holds  for  the  ?ith  terra  of  the  given 
series,  it  also  holds  for  the  (/i  +  l)st  term;  but  we  know 
tint  it  holds  for  the  fourth  term,  and  hence  it  also  holds 
for  the  fifth  term  ;  and  so  on. 


SUMMATION   OF   SERIES.  425 

Therefore  (1)  holds  for  any  term  of  the  given  series. 

Note.  If  the  differences  finally  become  zero,  the  value  of  «„  can  be 
obtained  exactly. 

590.    To  find  the  sum  of  the  first  n  terms  of  the  series 

tti,    a,,   a,,    a^,   as,    ... .  (1) 

Let  S  denote  the  sum  of  the  first  n  terms. 
Then  S  is  the  (w  +  l)st  term  of  the  series 

0,    ai, ■  Gi  +  a.,,   ai  +  a2  +  ttg,    •••  •  (2) 

The  first  order  of  differences  of  (2)  is  the  same  as  (1) ; 
whence  it  follows  that  the  rth  order  of  differences  of  (2)  is 
the  same  as  the  (r  —  l)st  order  of  differences  of  (1). 

If,  therefore,  cl^,  dj,  ...,  represent  the  first  terms  of  the 
1st,  2d,  ...,  orders  of  differences  of  (1),  a^  d^,  dj,  ...,  will  be 
the  first  terms  of  the  1st,  2d,  3d,  ...,  orders  of  differences 
of  (2). 

Putting  tti  =  0,  di  =  Oi,  do  =  d^,  etc.,  in  (3),  Art.  589,  we 
have 

^^^a,  +  ^^<^d,  +  ^^^^-V.^^^"^^^.+  --    (3) 

\A  \^ 

591.  Example.  Find  the  twelfth  terin  and  the  sum  of 
the  first  twelve  terms,  of  the  series  1,  8,  27,  64, 125,  .... 

Here,  n  =  12,  and  aj  =  1. 

Also,  di  =  7,  ^2  =  12,  c?3  =  6,  and  d,  =  Q)  (Art.  587) 

Substituting  in  (1),"  Art.  589,  the  twelfth  term 

=  1  +  11.7+^:^^.12  +  ^1-11^.6  =  1728. 
^        ^    1-2  1.2-3 

Substituting  in  (3),  Art.  590,  the  sum  of  the  first  twelve 

-12  +  12ill. 7  +  12.11-10. i2  +  l^-ll-lQ-9. 6  =  6084. 
~      ^    1-2        ^     1.2-3  1-2.3.4 


426  COLLEGE   ALGEBRA. 

592.  Piles  of  Shot. 

Example.  li  shot  are  piled  in  the  shape  of  a  pyramid 
with  a  triangular  base,  each  side  of  which  exhibits  9  shot, 
find  the  number  in  the  pile. 

The  number  of  shot  in  the  first  five  courses  are  1,  3,  6, 10, 
and  15,  respectively  ;  we  have  then  to  find  the  sum  of  the 
first  nine  terms  of  the  series  1,  3,  6,  10,  15,  .... 

The  successive  orders  of  differences  are  as  follows  : 

1st  order,  2,     3,     4,     5,     .... 

2d  order,  1,     1,     1,     .... 

3d  order,  0,     0,     .... 

Putting  Ti  =  9,  tti  =  1,  f?i  =  2,  and  ch  =  1  in  (3),  Art.  599, 
^  =  9  +  1^.2  +  1^.1  =  165. 

EXAMPLES. 

593.  1 .  Pind  the  first  term  of  the  sixth  order  of  differ- 
ences of  the  series  1,  3,  8,  20,  48,  112,  256,  .... 

2.  Find  the  eleventh  term,  and  the  sum  of  the  first  eleven 
terms  of  the  series  1,  8,  21,  40,  65,  .... 

3.  Find  the  ninth  term,  and  the  sum  of  the  first  nine 
terms  of  the  series  7,  14,  19,  22,  23,  ...  . 

4.  Find  the  thirteenth  term,  and  the  sum  of  the  first 
thirteen  terms  of  the  series  4,  14,  30,  52,  80,  .... 

5.  Find  the  sum  of  the  first  n  natural  numbers. 

6.  If  shot  are  piled  in  the  shape  of  a  pyramid  with  a 
square  base,  each  side  of  which  exhibits  31  shot,  find  the 
number  contained  in  the  pile. 

7.  Find  the  fourteenth  term,  and  the  sum  of  the  first 
fourteen  terms  of  the  series  8,  16,  0,  -64,  -200,  -432,  .... 


SUMMATION   OF   SERIES.  427 

8.  Find  the  sum  of  the  first  ten  terms  of  the  series  1, 
16,  81,  256,  625,  1296,  2401,  .... 

9.  If  shot  are  «piled  in  the  shape  of  a  pyramid  with  a 
triangular  base,  each  side  of  which  exhibits  n  shot,  find  the 
number  contained  in  the  pile. 

10.  Find  the  nth  term,  and  the  sum  of  the  first  n  terms 
of  the  series  1,  5,  12,  22,  35,  ...  . 

11.  How  many  shot  are  contained  in  a  pile  of  ten  courses 
whose  base  is  a  rectangle,  if  the  number  of  shot  in  the 
upper  course  is  15  ? 

12.  How  many  shot  are  contained  in  a  pile  of  n  courses 
whose  base  is  a  rectangle,  if  the  number  of  shot  in  the  upper 
course  is  m  ? 

13.  Find  the  eighth  term,  and  the  sum  of  the  first  eight 
terms  of  the  series  30,  144,  420,  960,  1890,  3360,  .... 

14.  Find  the  sum  of  the  squares  of  the  numbers  1,  2, . . .,  n. 

15.  Find  the  sum  of  the  cubes  of  the  numbers  1,  2,  ...,  n. 

16.  Find  the  7ith  term,  and  the  sum  of  the  first  n  terms 
of  the  series  1,  4,  10,  20,  35,  56,  ... . 

17.  How  many  shot  are  contained  in  a  truncated  pile  of 
seven  courses  whose  bases  are  rectangles,  if  the  numbers 
of  shot  in  the  length  and  breadth  of  the  upper  course  are  10 
and  6,  respectively  ? 

18.  How  many  shot  are  contained  in  a  truncated  pile  of 
n  courses  whose  bases  are  squares,  if  the  number  of  shot 
in  each  side  of  the  upper  base  is  m  ? 

IISTTERPOLATION. 

594.  Interpolation  is  the  process  of  introducing  between 
the  terms  of  a  series  other  terms  conforming  to  the  law  of 
the  series.  Its  usual  application  is  in  finding  intermediate 
numbers  between  those  given  in  Mathematical  Tables. 


428  COLLEGE   ALGEBRA, 

The  operation  may  be  effected  by  giving  fractional  values 
to  n  in  equation  (1),  Art.  589. 

595.    1.    Given  V5  =  2^361,  VG  =  2.4495,  V7  =  2.6458, 
V8  =  2.8284,  ... ;  find  V6.3. 

In  this  case  the  successive  orders  of  differences  are : 
.2134,     .1963,     .1826,     .... 
-.0171,  -.0137,    .... 
.0034,    .... 
Whence,  d^  =  .2134,  da  =  -  .0171,  d^  =  .0034,  .... 
Since  the  required  term  is  distant  1.3  intervals  from  V5, 
we  have  n  =  2.3. 

Substituting  in  (1),  Art.  589,  we  have,  approximately, 

VaB  =  2.2361  +  1.3  X  .2134  +  ^f^f(-M71) 
1x2 
1.3X.3X--.7      Q(j3^ 
1x2x3 
=  2.2361  +  .2774  -  .0033  -  .0002  =  2.5100. 


EXAMPLES. 

2.  Given  log  22  =  1.3424,  log  23  =  1.3617,  log  24  =  1.3802, 
log 25  =  1.3979,  ... ;  find  log 24.5. 

3.  Given  a/TO  =  4.12129,  -^tI  =  4.14082,  \/72  =  4.16017, 
...;  find  a/70.12. 

4.  The  reciprocal  of  22  is  .04545 ;  of  23,  .04348  ;  of  24, 
.04167  ;  etc.     What  is  the  reciprocal  of  22.8  ? 

5.  Given    log  109  =  2.03743,   log  110  =  2.04139,    log  111 
=  2.04532,  ...  ;  find  log  110.7. 

6.  Given  V37  =  6.08276,  V38  =  6.16441,  V39  =  6.24500, 
... ;  find  V37.48. 

7.  Given  log  11  =  1.04139,   log  12  =  1.07918,   log  13  = 
1.11394,  log  14  =  L14613,  ...  ;  find  log  13.28. 


DETEEMI^TANTS.  429 


XL.   DETERMINANTS. 

596.    Consider  the  equations 

a^x  +  biy  =  Ci, 

cioX  +  boy  =  C2. 
Solving,  we  obtain 

^  _  boCi  —  &1C2  _  CaCTi  —  Ciag 

The  common  denominator  may  be  written  in  the  form 


a„b,\^  W 

which  is  understood  as  signifying  the  product  of  the  upper 
left-hand  and  lower  -right-hand  quantities,  minus  the  prod- 
uct of  the  lower  left-hand  and  upper  right-hand. 

The  expression  (1)  is  called  a  Determinant  of  the  Second 
Order. 

597.   The  numerators  of  the  fractions  in  the  preceding 
article  can  also  be  expressed  as  determinants  ;  thus. 


6.c,-6,C2=    ll  ^^  ,andc2a,- 

—  Cjtta  = 

"2, 

C2 

598.   Consider  the  equations 

a^x  +  b^y  -f  CiZ  = 

■ch 

a^  ■\-  boy  -\-  c^z  = 

:d2, 

a^x  +  63?/  +  CqZ  = 

■■d,. 

Solving,  we  obtain 

^  _  diboCs  —  di^sCo  +  dobgCi  —  chb 

1C3  +  d^h^ 

Cg  — 

■  dAci 

0'lb2Cs  —  C(,lbsC2  +  (i^^sCl  —  tta^l^S  +  «3&lC2  —  a3&2Ci 

with  results  of  similar  form  for  y  and  z. 


(1) 


430 


COLLEGE   ALGEBRA. 


The  denominator  of  (1)  may  be  written  in  tlie  form 


ttl, 

K 

Ci 

«2, 

b,, 

Co 

ttg, 

h, 

Cg 

(2) 


which  is  understood  as  signifying  the  sum  of  the  prod- 
ucts of  the  quantities  connected  by  lines  parallel  to  a  line 
joining  the  upper  left-hand  corner  to  the  lower  right-hand, 
in  the  following  diagram,  minus  the  sum  of  the  products 
of  the  quantities  connected  by  lines  parallel  to  a  line  join- 
ing the  lower  left-hand  corner  to  the  upper  right-hand. 


The  expression  (2)  is  called  a  Determinant  of  the  Third 
Order. 

599.   The   numerator    of    the   value    of   x  can   also   be 
expressed  as  a  determinant,  as  follows  : 


f?l, 

K 

Ci 

ch, 

b,, 

Ca 

ds, 

bs, 

Cs 

as  may  be  verified  by  expanding  it  by  the  rule  of  Art.  598. 


EXAMPLES. 
600.   Evaluate  the  following  determinants 


2,  3,  5 

10,  2,  8 

-2,  -3,  4 

7,  1,  4 

2. 

n,  4,  0 

3. 

5,  -0,  7 

6,  2,  3 

3,  1,  7 

-8,      9,  1 

DETERMINANTS. 


431 


1,  a,   b 

a,  h;  g 

1,  b,   c 

5. 

h,  b,  f 

1,   c,  a 

9,  f,   c 

1, 

c, 

-b 

— c, 

1, 

a 

b, 

-a, 

1 

Verify  the  following  by  expanding  the  determinants 


7. 


8. 


ag,      63,      C3 

Ct'25     0^5     Co 
"3;    ^3J    C3 


-  n    I  ^2,    Co  I    ,    ;  J  C2,    0^2  I    ,    .  I    CU,    b., 
I    '^SJ     *^3   I  I    ^^S)     ""3   I  I     "3?     ^^3 


«1, 

ao, 

«3 

bu 

b,, 

h 

Cl, 

C2, 

C3 

61,  ai,  Cl 

b.2,     0,-2)     C2 
^35     <^3>    ^^3 


10, 


«1, 

tti, 

&I 

a.,, 

«2, 

&., 

^3, 

«3, 

&3 

0.       11. 


mai,  61,  Cl 

ma2,  &2,  C2 

=  m 

mas,  h,  C3 

«i, 

K 

Cl 

ao, 

h, 

C2 

as, 

h, 

C3 

601.  General  Definition  of  a  Determinant. 

If,  in  any  permutation  of  the  numbers  1,  2,  3,  ...,  n,  a 
greater  number  precedes  a  less,  there  is  said  to  be  an 
inversion. 

Thus,  in  the  case  of  five  numbers,  the  permutation  4,  3, 
1,  5,  2,  has  six  inversions ;  4  before  1,  3  before  1,  4  before  2, 
3  before  2,  5  before  2,  and  4  before  3. 

Consider,  now,  the  n^  quantities 


^n,  I,    ^n^  2,    a„^  2 


a„ 


(1) 


Note  1.  The  notation  in  regard  to  suffixes,  in  tlie  above,  is  tliat 
the  first  suffix  denotes  tlie  liorizontal  row,  and  tlie  second  the  vertical 
column,  in  which  the  quantity  is  situated. 

Thus,  aii,r  is  the  quantity  in  the  klh  row  and  rth  column. 


432  COLLEGE   ALGEBRA. 

Let  all  possible  products  of  the  quantities  taken  n  at  a 
time  be  formed,  subject  to  the  restriction  tliat  each  product 
shall  contain  one  and  only  one  quantity  from  each  row,  and 
one  and  only  one  from  each  column,  and  write  them  so  that 
the  second  suffixes  shall  occur  in  the  order  1,  2,  ...,  n. 

Note  2.  This  is  equivalent  to  wi'iting  all  the  permutations  of  the 
order  1,  2,  ...,  n  in  the  first  suffixes. 

Give  to  each  product  the  sign  +  or  —  according  as  the 
number  of  inversions  in  the  first  suffixes  is  even  or  odd. 

The  expression  (1)  is  called  a  Determinant  of  the  nth 
Order. 

602.  The  expanded  form  may  also  be  obtained  by  writing 
the  Jirst  suffixes  in  the  order  1,  2,  ...,  n,  and  giving  to  each 
product  the  sign  +  or  —  according  as  the  number  of  inver- 
sions in  the  second  suffixes  is  even  or  odd. 

For  let  the  absolute  value  of  any  product,  obtained  as  in 
Art.  601,  be 

ap,ia,,2 -"^'r.n;  (1) 

where  jy,  q,  ...,  r  is  a  permutation  of  1,  2,  ...,  n. 

Writing  the  first  suffixes  in  the  order  1,  2,  ...,  n,  we  have 

«1,8  «2,«    •••  C*K,t;)  (2) 

where  s,  i,  ...,  ^;  is  a  permutation  of  1,  2,  ...,  n.    ■ 

It  is  evident  that  there  are  just  as  many  inversions  in 
the  first  suffixes  of  (1)  as  in  the  second  suffixes  of  (2) ;  and 
hence  the  products  (1)  and  (2)  will  have  the  same  sign. 

603.  The  quantities  aj^i,  Oj^s,  etc.,  are  called  the  constitu- 
ents of  the  determinant,  and  the  products  aj,]  aj^j  •••  ^n^M  etc., 
occurring  in  the  expanded  form,  are  called  its  elements. 

The  constituents  lying  in  the  diagonal  joining  the  upper 
left-hand  corner  to  the  lower  right-hand,  are  said  to  be  in 
the  principal  diagonal;  the  element  whose  factors  are  the 
constituents  in  the  principal  diagonal  is  always  positive. 

Note.  By  Art.  .54.3,  the  number  of  elements  in  the  expanded  form 
of  a  determinant  of  the  nth  order  is  |  n.  , 


DETERMINANTS. 


433 


604.  It  may  be  shown  that  the  definition  of  Art.  601 
agrees  with  those  of  Arts.  596  and  598  ;  for  consider  the 
determinant 

^i,i>  %,25  <^i,a 

0^2,  U     <^2,2>     C(2  3 
^S,  1}     %,  2?     '^3,  a 

The  only  possible  products  of  the  quantities  taken  3  at  a 
time,  subject  to  the  restriction  that  each  product  shall  con- 
tain one  and  only  one  constituent  from  each  row,  and  one 
and  only  one  from  each  column,  the  second  sufl&xes  being 
written  in  the  order  1,  2,  3,  are 

%,  1  <^2,  2  <^3,  3)     ^1,1^,2^2,35     ^2, 1  <^1,  2  <^3,  3?    <^2, 1  %,  2  <^1, 3) 
%1<^1,2^2,3J    ^^^    %,  1  f'2,  2  %,  3- 

In  the  first  of  these  there  are  no  inversions  in  the  first 
suffixes  ;  in  the  second  there  is  one,  3  before  2 ;  in  the  third 
there  is  one ;  in  the  fourth,  two  ;  in  the  fifth,  two  ;  in  the 
sixth,  three. 

Then  according  to  the  rule  of  Art.  601,  the  first,  fourth, 
and  fifth  products  are  positive,  and  the  second,  third,  and 
sixth  are  negative  ;  and  the  expanded  form  is 

«1, 1  «2,  2  ^3,  3  —  «1, 1  ^3,  2  <^2,  3  —  «2, 1  «1,  2  «3, 3  +  ^2, 1  %,  2  «1, 3 
+  «3, 1  Ctl,  2  «2,  3  —  «3, 1  «2,  2  «1,  3) 

which  agrees  with  Art.  598. 


PROPERTIES  OF  DETERMINANTS. 

605.   A  determinant  is  not  altered  in  value  if  its  roios  are 
changed  to  columns,  and  its  columns  to  rows. 
(Compare  Ex.  8,  Art.  600.) 
Consider  the  determinants 


<-"!,  1>    "'1,2) 
•^'2,15  .'^'2,  2j 


and 


''M,l;    '-''2,1) 
a^i  2,     ^^2  2) 


434 


COLLEGE   ALGEBRA. 


Since  the  first  suffixes  of  the  first  determinant  are  the 
same  as  the  second  suffixes  of  the  second,  if  the  first  deter- 
minant is  expanded  by  the  rule  of  Art.  601,  and  the  second 
by  the  rule  of  Art.  602,  the  results  will  be  the  same. 

Therefore  the  determinants  are  equal. 

606.   A  determinant  is  changed  in  sign  if  any  ttvo  consecu- 
tive columns,  or  any  tivo  consecutive  rows,  are  interchanged. 
(Compare  Ex.  9,  Art.  600.) 
Consider  the  determinants 


■n,«5    a«,r, 


a. 


and 


Cf2,lJ    '••)    ^2,r1     <^2,  }J    »•■>    tt2,; 


the  gth  and  rth  columns  of  the  first  being,  respectively,  the 
rth  and  gth  columns  of  the  second. 

Let  the  absolute  value  of  one  of  the  elements  of  the  first 
determinant  be 

a,,i...a(,a„,,...a,,„;  (1) 

where  s,  ...,  t,  u,  ...,  v  is  a  permutation  of  1,  2,  ...,  n. 

Since  the  constituent  in  the  ith  row  and  gth  column  of 
the  second  determinant  is  a<_^,  and  the  constituent  in  the 
?*th  row  and  rth  column  a„^  „  the  absolute  value  of  the  cor- 
responding element  of  the  second  determinant  may  be 
derived  from  (1)  by  replacing  a,,,  and  a,,,^  l^y  «(,r  and  a„,,, 
respectively  ;  that  is, 

a^^  1 . . .  a<_  ^  a„_ , . . .  a„_  „. 

The  latter  expression  is  also  the  absolute  value  of  one  of 
the  elements  of  the  first  determinant,  since  it  has  one  and 
only  one  constituent  from  each  row,  and  one  and  only  one 
from  each  column  ;  and  writing  it  so  that  the  second  suf- 
fixes shall  occur  in  the  order  1,  2,  ...,  ?;,  wc  have . 


(2) 


DETERMINANTS.  435 

Now  whatever  the  number  of  inversions  in  the  first  suf- 
fixes of  (1),  s,  ...,  t,  u,  ...,  V,  tlie  number  of  inversions  in 
the  first  suffixes  of  (2),  s,  ...,  u,  t,  ...,  v,  differs  from  it  by 
unity  ;  for  in  the  first  case  t  precedes  u,  and  in  the  second 
u  precedes  t. 

Hence  the  elements  (1)  and  (2)  of  the  first  determinant 
are  of  opposite  sign  (Art.  601). 

That  is,  any  two  elements  of  the  given  determinants  of 
equal  absolute  value  are  of  opposite  sign;  and  hence  the 
determinants  themselves  are  of  equal  absolute  value  and 
opposite  sign. 

It  follows  from  Arts.  605  and  606  that  if  two  consecu- 
tive rows  are  interchanged,  the  sign  of  the  determinant  is 
changed. 

607.  A  determinant  is  changed  in  sign  if  any  two  roics,  or 
any  two  columns,  are  interchanged. 

Consider  the  m  letters  a,  b,  c,  .. .,  e,  f,  g. 

By  interchanging  a  with  b,  then  a  with  c,  and  so  on  in 
succession  with  each  of  the  m  —  1  letters  to  the  right  of  a, 
a  may  be  brought  to  the  right  of  g. 

Then,  by  interchanging  g  with  f,  then  g  with  e,  and  so  on 
in  succession  with  each  of  the  m  —  2  letters  to  the  left  of  g, 
g  may  be  brought  to  the  left  of  b. 

It  is  evident  from  this  that  a  and  g  may  be  interchanged 
by  (m  —  1)  -t-  (m  —  2),  or  2m  —  3,  interchanges  of  consecu- 
tive letters  ;  that  is,  by  an  odd  number  of  interchanges  of 
consecutive  letters. 

It  follows  from  the  above  that  any  two  rows,  or  any  two 
columns,  of  a  determinant  may  be  interchanged  by  an  odd 
number  of  interchanges  of  consecutive  rows  or  columns. 

But  every  interchange  of  two  consecutive  rows  or  columns 
changes  the  sign  of  the  determinant  (Art.  606). 

Therefore  the  sign  of  the  determinant  is  changed  if  any 
two  rows,  or  any  two  columns,  are  interchanged. 


436 


COLLEGE   ALGEBRA. 


608.  If  two  rows,  or  two  columns,  of  a  determinant  are 
identical,  the  value  of  the  determinant  is  zero. 

(Compare  Ex.  10,  Art.  600.) 

Let  D  be  the  value  of  a  determinant  having  two  rows,  or 
two  columns,  identical. 

If  these  rows,  or  cohimns,  are  interchanged,  the  value  of 
the  resulting  determinant  is  —  7)  (Art.  607). 

But  since  the  rows,  or  columns,  which  are  interchanged 
are  identical,  the  two  determinants  are  of  equal  value. 

Hence,  D  =  —  D ;  and  therefore  D  =  0. 


609.    Cyclical  Interchange  of  Rows  or  Columns. 

By  7i  —  1  successive  interchanges  of  two  consecutive  rows, 
the  upper  row  of  a  determinant  of  the  nth  order  may  be 
brought  to  the  bottom. 

Thus,  by  Art.  606,  the  determinant 


is  equal  to  (— 1)"  ' 


as,  &2J  •• 


a„,  5„, 


The  above  is  called  a  cyclical  interchange  of  rows. 

In  like  manner,  by  n  —  1  successive  interchanges  of  two 
consecutive  columns,  the  left-hand  column  of  a  determinant 
of  the  ?ith  order  may  be  brought  to  the  end. 

610.  If  each  constituent  in  one  roiv,  or  in  one  column,  is 
the  sum  of  m  terms,  the  determinant  can  be  expressed  as  the 
sum  ofm  determinants. 

Consider  the  determinant 


^"1,  15 
«2,1> 


«■«  n 


(1) 


DETERMINANTS. 


437 


Let  each  constituent  in  the  rth  column  be  the  sum  of  7n 
terms,  as  follows  : 

ai,r  =fh  fci-i \-fi, 

C('2,r  =  &2  +  C2   H +/2, 


&„  +  c„ +  •••+/„. 


Let  ap,i'"a,,^---a^^„  be  the  absolute  value  of  one  of  the 
elements  of  (1) ;  then, 

=  (a^,,...6,...a,,„)+...  +  (a,.i.../,...a,,„). 

It  is  evident  from  this  that  the  determinant  (1)  can  be 
expressed  as  the  sum  of  the  determinants 


ttl 

1}  • 

,  b„  . 

;    «l,n 

tti 

1)   • 

;  fv    . 

.,     tti 

n 

a2 

1)  • 

V  b,,  . 

•,     «2,n 

+  • 

•  + 

a„ 

I'  .* 

•,  J2,    . 

.,   fn,     . 

n 

a, 

,  1'  • 

.,  K  . 

•,     «„,„ 

n 

611.  If  all  the  constihients  in  one  row,  or  in  one  column, 
are  multiplied  by  the  same  quantity,  the  determinant  is  multi- 
plied by  this  quantity. 

Consider  the  determinant 


•,  ai,r, 


.,  a„  , 


(1) 


Multiplying  each  constituent  in  the  rth  column  by  m,  we 
have 


ti,i,  ...,  maj,r, 


ma. 


2,r)      '••■>     ^'2,1 


6„,„  ...,  ma^^,, 


(2) 


Let  Op  1 . . .  a, ,. . . .  a,,  „  be  the  absolute  value  of  one  of  the 
elements  of  (1). 


438 


COLLEGE  ALGEBRA. 


Eeplacing  a^^^  by  ma^^^,  the  absolute  value  of  the  corre- 
sponding element  of  (2)  is  wia^_  i . . .  a,_  ^ . . .  o,_  „. 

It  is  evident  from  this  that  the  determinant  (2)  is  equal 
to  the  determinant  (1)  multiplied  by  m. 


612.  If  the  constituents  in  any  row,  or  column,  are  multi- 
plied by  the  same  quantity,  and  either  added  to,  or  subtracted 
from  the  corresponding  constituents  of  another  row,  or  column, 
the  value  of  the  determinant  is  not  changed. 

Let  the  constituents  of  the  rth  column  of  the  following 
determinant  be  multiplied  by  m,  and  added  to  the  corre- 
sponding constituents  of  the  gth  column. 


,   "-!>     •••>     "'gi     '••)     "-rj     •••5     "^11 

We  then  obtain  the  determinant 


(1) 


a,  +  ma^, 
b^  +  7nb„ 


a„  .. 
h,,  .. 


ki,  ...,  Jc^  +  mk^,  ...,  k^,  ...,  kn 

rhich,  by  Arts.  610  and  Gil,  is  equal  to 


(2) 


bu  ...,   b„ 


.,  a^. 


ki, 


•1     l^rj 


k„ 


+  m 


b„  ...,  K, 
ki,  ...,  k^, 


.,  a,, 

.,     br. 


But  the  coefficient  of  m  is  equal  to  zero  (Art.  608). 
Hence  the  determinant  (2)  is  equal  to  (1). 


.,  a„ 


613.   Minors. 

If  the  constituents  in  any  m  rows  and  any  m  columns  of 
a  determinant  of  the  nth  order  are  erased,  the  remaining 
constituents  form  a  determinant  of  the  (n— m)th  order. 


DETERMINANTS. 


439 


This  determinant  is  called  an  mth  Minor  of  the  given 


determinant  5  thus, 


a„ 

d„ 

6] 

%, 

d„ 

es 

a« 

ds, 

Go 

is  a  second  minor  of 


«!,  6i,  Ci,  c?i,  ei 

<*2)  O2,  C2,  tto,  62 

0^3)  "3j  ^3?  ^3j  63 

(X45  0^  C4J  Ct4,  64 

%,  &s,  Co,  d„  65 


obtained  by  erasing  the  second  and  fourth  rows,  and  the 
second  and  third  columns. 


614.    To  find  the  coefficient  of  a^^-^  in  the  determinant 


«2,1J     «2,2>     •• 

;     «1, 
',     f2, 

«n.b    «n.2)     • 

.,  a,. 

(1) 


By  Art.  601,  the  absolute  values  of  the  elements  which 
involve  a^^i  are  obtained  by  forming  all  possible  products  of 
the  constituents  taken  71  at  a  time,  subject  to  the  restric- 
tions that  the  first  constituent  shall  be  aj^i,  and  that  each 
product  shall  contain  one  and  only  one  constituent  from 
each  row  except  the  first,  and  one  and  only  one  from  each 
column  except  the  first. 

It  is  evident  from  this  that  the  coefficient  of  a^^i  in  (1) 
may  be  obtained  by  forming  all  possible  products  of  the 
following  constituents  taken  n  —  1  at  a  time, 

%25    f'3,3>     •••?    0.3,  n 


a«,2,    «„,3J    •••,    ««,» 

subject  to  the  restriction  that  each  product  shall  contain 
one  and  only  one  constituent  from  each  row,  and  one  and 
only  one  from  each  column,  writing  the  second  suffixes  in 
the  order  2,  3,  ...,  n,  and  giving  to  each  product  the  sign  + 
or  —  according  as  the  number  of  inversions  in  the  first 
suffixes  is  even  or  odd. 


440  COLLEGE   ALGEBRA. 

Then  by  Art.  601,  the  coefficient  of  aj,  i  is 
a,,,,  a,, 3,   ...,  a.„ 


that  is,  the  minor  obtained  by  erasing  the  first  row  and  the 
first  column  of  the  given  determinant. 

615.  By  aid  of  the  theorem  of  Art.  614,  a  determinant  of 
any  order  may  be  expressed  as  a  determinant  of  any  higher 
order  ;  thus, 

1,  0,    0,   0,         1'  ^'  0'    0,   0 

0,  ai,  b„  Ci 

0,  a.2,  b.2,  c, 

0,  as,  63,  Cg 


oi,  bi,  ci 

as,  &2,  C2 

= 

as,  &3;  C3 

0,  1,  0,  0,  0 

0,  0,  ai,  bi,  ci 

0,  0,  a.2,  62?  ^2 

0,  0,  aa,  63,  C3 


etc. 


616.  We  will  now  consider  the  general  case. 
To  find  the  coefficient  of  a^^^  in  the  determinant 
ai.b    ••;   a,,,    ...,   aj 


(1) 


By  fc  —  1  successive  interchanges  of  consecutive  rows, 
and  r—1  successive  interchanges  of  consecutive  columns, 
the  constituent  a^ ,.  may  be  brought  to  the  upper  left-hand 
corner;  thus,  by  Art.  606,  the  determinant  is  equal  to 


(-iy\-iy-^ 


ai,r;  ai,b 


a*  „ 


Then  by  Art.  614,  the  coefficient  of  a^^^  is 
«!,!,   •••,  a, 


(-1)''+'-^ 


But  (-1)^+'-  2  =  ( -  l)*"-( -  1) ^  =  ( -  1  )*+'•. 


DETERMmANTS. 


441 


Hence  the  coefficient  of  a^^^  is  equal  to  (—1)*+',  multi- 
plied by  tliat  minor  of  (1)  Avhich  is  obtained  by  erasing  the 
^th  row  and  ?-th  column, 

617.  By  aid  of  Art.  616,  a  determinant  of  any  order  may 
be  expressed  in  terms  of  determinants  of  any  lower  order. 

Thus,  since  every  element  of  a  determinant  contains  one 
and  only  one  element  from  the  first  row,  we  have, 

tti,  &i,  Ci,  di 

a.,,  bo,  Co,  cZa 

<^3;    ^g,    Cg,    ttg 

a^,  &4,  C4,  di 


=  «! 

b.,,  c.2,  d.2 

b-i,    Cg,    dg 

&4,  C4,  di 

-by 

a.2,  C2,  d.2 

Cls,    Cg,    dg 

(li,  C4,  d, 

+  Ci 

a.2,  &2,  di 

ttg,     63,    dg 

a^,  bi,  di 

-  dl      ttg,    63,    Cg 

5 

cii,  bi,  C4 

and  each  of  the  latter  determinants  may  in  turn  be  expressed 
in  terms  of  determinants  of  the  second  order. 

618.   Evaluation  of  Determinants. 

The  method  of  Art.  617  may  be  used  to  express  a  deter- 
minant of  any  order  higher  than  the  third  in  terms  of 
determinants  of  the  third  order,  which  may  then  be  evalu- 
ated by  the  rule  of  Art.  598. 

The  theorem  of  Art.  612  may  often  be  advantageously 
employed  to  shorten  the  process,  as  shown  in  Ex.  1. 

5,  7,  8,  6 

11,  16,  13,  11 

14,  24,  20,  23 

7,  13,  12,  2 


1 .    Evaluate 


Subtracting  the  first  row  from  the  last,  twice  the  first 
row  from  the  second,  and  three  times  the  first  row  from  the 
third  (Art.  612),  the  determinant  becomes 


,  by  Art.  61L 


5,  7, 

^, 

6 

5.  7, 

8, 

6 

1,  2, 
1,  3, 

-3, 

-4, 

-1 
5 

=  2 

1,  2, 
-1,  3, 

-3, 
-4, 

-1 
5 

2,  6. 

4, 

-4 

1,  3, 

9 

—  2 

442 


COLLEGE   ALGEBRA. 


3, 

23, 

11 

5, 

-7, 

4 

,  by  Art.  616. 

1, 

5, 

-1 

Subtracting  five  times  the  second  row  from  the  first,  add- 
ing the  second  row  to  the  third,  and  subtracting  the  second 
row  from  the  last,  we  have 


0,  -3,  23,  11 

1,  2,  -3,  -1 

0,  5,  -7,  4 

0,  1,  5,  -1 


The  object  of  the  above  process  is  to  put  the  given  deter- 
minant in  such  a  form  that  all  but  one  of  the  constituents  in 
one  column  shall  be  equal  to  zero ;  the  determinant  can  then 
be  expressed  as  a  determinant  of  the  third  order  by  Art.  616. 

The  last  determinant  may  be  evaluated  by  Art.  598  ;  but 
it  is  better  to  subtract  five  times  the  first  column  from  the 
second,  and  then  add  the  first  column  to  the  last :  thus, 


38,  8 

■32,  9 

0,  0 


=  -2 


■32   9!= -2(342+256)  =  -1196. 


EXAMPLES. 


Evaluate  the  following 


3. 


5,  15,  10 

6,  21,  13 

7,  25,  16 

6. 

11,  12,  13 
14,  15,  16 
17,  18,  19 

•  7. 

30,  15,  17 

29,  18,  23 
20,  19,  22 

.  8. 

1,  a,  a- 
1,  b,   h- 

9. 

1,  c,  c- 

1,  2,  3,  4 

1,  3,  6,  10 

10. 

1,  4,  10,  20 

1,  5,  15,  35 

9,  13,  17,  4 

18,  28,  33,  8 

30,  40,  54,  13 

24,  37,  46,  11 


13. 


1,  15,  14,  4 

12,  6,  7,  9 
8,  10,  11,  5 

13,  3,  2,  16 


7,  13,  10,  6 

5,  9,  7,  4 

11. 

8,  12,  11,  7 

4,  10,  6,  3 

a,   1, 

1,  1 

1,  h, 

1,  1 

1,  1, 

c,  1 

1,  1, 

1,  d 

3,  2,  1,  4 

15,  29,  2,  14 

•  12. 

16,  19,  3,  17 

33,  39,  8,  38 

0, 

a, 

b, 

c 

a, 

0, 

c, 

b 

b, 

c, 

^, 

a 

c, 

b> 

a, 

0 

a,  b,  c,  d 

b,  a,  d,  c 

c,  d,  a,  b 

d,  c,  b,  a 


DETERMINANTS. 


443 


14. 


15. 


18. 


X  —  4:y,  x  —  y,  x  +  2y 
x  —  oy,  X,  X  +  'dy 

x-2y,  x  +  y,  a; +  4?/ 


x-i-  y,  z  —  y,  z  —  X 
x  —  y,  y  +  z,  x  —  z 
y  —  x,  y-z,  z  +  x 

p,       q,       r,    s 
~p,       q,       r,    s 

-P,   -Q,       ^»    « 
—  p,   —q,   —r,    s 


19. 


16. 

7,' 
—2 

5, 

-2,      0,  5 

6,   -2,  2 

-2,      5,  3 

2,       3,  4 

0, 

a,      b,   c 

17. 

—  a, 

0,      11,  VI 
-n,      0,    I 

-c, 

-m,  -  I,  0 

a-{-x, 

a, 
a, 

a,  ^ 

b+x, 
b, 
b, 

c,         d 
c,         d 
c+x,      d 
c,      d+x 

619.   Let  A^^r  denote  the  coefficient  of  a^^^  in  tlie  deter- 
minant 

a,  1 «!  ..,   ...,  a, 


a^ 


(1> 


Then  since  every  element  of  the  determinant  involve? 
one  and  only  one  constituent  from  the  rth  column,  the  valu^ 
of  the  determinant  is 

It  follows  from  the  above  that 

is  the  value  of  a  determinant  which  differs  from  (1)  only  in 
having  b^  b.,,  ...,  b„  instead  of  ai_^,  03,^?  •••;  ««,rj  as  the  con- 
stituents in  the  rth  column. 

Hence,  if  q  is  any  number  of  the  series  1,  2,  ...,  w  except 
r,  the  expression 

for  it  is  the  value  of  a  determinant  whose  qth.  and  rth  col- 
umns are  identical  (Art.  608). 


444 


COLLEGE   ALGEBRA; 


620.   Multiplication  of  Determinants. 

Consider  the  determinant 

ttidi  +  a,ei  +  tts/i,  bidi  +  62^1  +  hfi,  Cid^  +  c^ei  +  cj^ 
a^do  +  a^e^  +  03/2,  61^2  +  6262  +  hU  c^d^  +  0263  +  C3/2 
ttjcZg  +  02^3  +  a 3/3,  bids  +  6263  +  &3/3J  Cjdg  +  Cgeg  +  C3/3 


■    By  Art.  610,  this  can  be  expressed  as  the  sum  of  twenty- 
seven  determinants,  of  which  the  following  are  types  : 


aA,  he^,  C3/1 

a^,  b-^So,  Csf, 

, 

aA,  b^es,  C3/3 

ttidi,  bA,  Cidi 
aA,  ^Aj,  CA2 
aA,  bA}  cA 


That  is,  by  Art.  611, 


aib2C3 


d„  ei,  /i 

do,   62,  /a 

,    aAc2 

ch,  es,  f-i 

(h,  di,  e, 

C?2,     d.2,     6.2 

,    ciAci 

ds,  cl„  e.. 

d„ 

d„ 

di 

d,, 

d,, 

d. 

ds, 

d„ 

d, 

The  eighteen  determinants  of  the  second  type,  and  the 
three  determinants  of  the  third  type,  are  all  equal  to  zero 
by  Art.  608. 

Hence  the  given  determinant  is  equal  to 


aAcs 


H-aa&sCj 


di,  61,  /i 
d,,  e,,  /a 
d^j  €3,  fs 


■j-aAc2 


di,  fi,  d^ 

^2,  fi,  d^   +a3^iC2 

63?  fi,  di 


di,  fi,  61 
d,,  f,  e, 
d^,  f,  63 

+(hbjC3 

A,  di,  e, 
f,  d,,  62 
f,  d^,  63 

4-a3&2Ci 

eij 

d„ 

f 

62, 

d,, 

f 

^3, 

d3. 

f 

f. 

Ci, 

di 

f, 

e,, 

d. 

f, 

es, 

d3 

By  Art.  606,  the  above  is  equal  to 

{ciACs — aAc2  +  <^^A<^i  -  aa&iCs  +  cta^iCs  -  ctAci) 


or, 


ay, 

&1, 

Ci 

«2, 

b,, 

C-2 

X 

(hi 

h, 

Cy 

di,  e.,,  f, 
dz,  Co,  f 


d„ 

Ci, 

/i 

d„ 

62, 

/2 

d3. 

63, 

f 

That  is,  the  product  of  two  determinants  of  the  third 
order  can  be  expressed  as  a  determinant  of  the  third  order. 


DETERMINANTS. 


445 


621.   We  will  now  discuss  the  general  case. 
Consider  the  determinants 


,  and  Q  ■■ 


,  K 


R  = 


(1) 


(2) 


Let  a  third  determinant 
be  formed  from  P  and  Q  by  aid  of  the  relation 

Ck,  r  =  tti,  1  &r,  1  +  «i,  2  ^r,  2  H h  «l,  „  ^r, ; 

To  prove  that  R  =  Px  Q. 

One  of  the  elements  of  i2  is  ±  Cp_i...Cg,„,  where  j9,  ...,  q  is 
a  permutation  of  1,  2,  ...,  7i ;  the  sign  being  +  or  —  accord- 
ing as  the  number  of  inversions  in  j?,  ...,  g  is  even  or  odd. 

By  (2),  the  value  of  this  element  is 

±  (%,  1  ^1, 1  +  •  •  •  +  «>, nKn)  —  (S.  1  ^«, !+•••+  «*, « ^«, ») ■ 

Expanding,  we  obtain  a  series  of  terms  of  the  type 

±(cv,...a„,)x  (&,,.•..&.,.);  (3) 

where  s,  ...,  t  are  numbers  of  the  series  1,  2,  ...,  n. 

Then  R  is  equal  to  the  sum  of  all  the  terms  of  which  (3) 
is  the  type ;  the  sign  of  each  being  +  or  —  according  as 
the  number  of  inversions  in  p,  ...,  q  is  even  or  odd. 

Now  since p,  ...,  g  is  a  permutation  of  1,  2,  ...,  n,  by  Art. 
601  the  sum  of  all  terms  involving  6i, , . . .  &„, ,  is 


X  &],.•••&„,«! 


(4) 


and  this  expression  vanishes  identically  unless  s, 
permutation  of  \,  2,  ...,  n  (Art.  60S). 


446 


COLLEGE   ALGEBRA. 


But  if  s,  ...,  t  is  a,  permutation  of  1,  2,  ...,  n,  (4)  may  be 
written 

o,  1,   ....  o, 


a„ 


X  &],, 


(5) 


the  sign  being  +  or  —  according  as  the  number  of  inver- 
sions in  s,  ...,  t  is  even  or  odd  (Art.  607). 

Hence  the  sum  of  all  the  terms  of  which  (5)  is  the  type  is 


«n,l5     •••5     «n 

Therefore  E=Px  Q. 

622.  It  was  shown  in  Art.  621  that  the  product  of  two 
determinants  of  the  nth  order  can  be  expressed  as  a  deter- 
minant of  the  ?ith  order. 

But  a  determinant  of  any  order  can  be  expressed  as  a 
determinant  of  any  higher  order  (Art.  615). 

Hence,  the  product  of  any  two  determinants  can  be 
expressed  as  a  determinant  of  the  same  order  as  that  of 
the  factor  of  highest  order. 

623.  Application  to  the  Solution  of  Equations. 

Let  it  be  required  to  solve  the  following  set  of  n  indepen- 
dent, simultaneous  simple  equations,  involving  n  unknown 
quantities  : 

«2, 1  ^1  +  Cf L',  2  ^'2  +  •  •  •  +  «2,  n^n  =  ^2) 


Let  Ai,^r  denote  the  coefficient  of  a^.^^  in  the  determinant 


Z)  = 


"1,1?     "1,2?     •• 

a..^,  «„  2,  ■• 


DETERMINANTS. 


447 


Multiplying  the  given  equations  by  A-^^^,  A^^^ 
respectively,  and  adding  the  results,  we  have 

+  X^  (tti, ,.  ^1,  r  +  «2,  r^2,r-\ +  «„,  r  -4„,  r)  + 

=  6l  A,,-  +        &2  Ar  H h      K^n,r- 


a) 


By  Art.  619,  the  coefficient  of  each  of  the  unknown  quan- 
tities except  XV  is  equal  to  zero ;  also,  the  coefficient  of  x^ 
is  D,  and  the  second  member  is  a  determinant  which  differs 
from  D  only  in  having  &i,  62)  •••?  ^n  i^  place  of  aj^^,  aa,^,  ...,  a„,r 
as  the  constituents  of  the  ?-th  column. 

Denoting  the  latter  by  D^,  we  have 

x.  =  ^- 
D    ^ 

Example.   Find  the  value  of  .?/  from  the  equations 

3x-5y  +  7z=  28, 
2x  +  6y-9z  =  -23, 
4.x-2y-5z=       9. 


We  have     y  = 


3, 

28, 

7 

2, 

-23, 

-9 

4, 

9, 

-5 

3, 

-5, 

7 

2 

6, 

-9 

4, 

0 

■"5 

-5 

630 


210 


=  -3. 


624.  Elimination. 

Consider  the  following  n  homogeneous  simple  equations, 
involving  n  unknown  quantities  : 

ttg  iXi  -\-  a2,2-'^'2  +  •••   +  02,n^n  =  0, 


<^»t,l^'l  +  f*n,2^'2  +•••+«« 


0. 


448 


COLLEGE   ALGEBRA. 


Dividing  the  terms  of  each  equation  by  x„,  and  transpos- 
ing, we  have 


ai,i^  +  «i,2^  +  •••  +  «i,n-i^ 


«!,, 


(1) 


Xi  Xo  Xn-i 

a2,l  ;^  +  «2,2  r^  +  •••  +  ^.n-l^T"  =  -  ^2.' 


Xi  X2  X„_i 

By  solving  the  last  n  —  1  equations,  we  may  obtain  the 
values  of  the  n  —  1  quantities  —  >  •••>  -^^ ;  thus,  by  Art.  623, 


—  «2,n>     «2,2J     • 

•,     «2,„-I 

—  a„,n>    «n,2,     • 

V    «n,H-l 

C^2,b     «2,2,     •• 

•,     «2,„-l 

««,!,    Ct„,2,     . 

V  a„,„_i 

a2,2,    ■■}  a2,n-l,   Hn 

(_!)«-! 

Cb,^2i   •••)  Cln,n-\^   ^n,n 

%,1)    01.2, 2J    •••}    <^2,n-l 

a«,i,  a„,2,  ...,  a„,„_i 

by  Art.  606 ;  and  results  of  similar  form  will  be  found  for 
X.2  cc„_i 

xj       '    Xn 

Substituting  these  values  in  (1),  clearing  of  fractions, 


and  transposing,  we  have 

av,2)    •••)   Ct2,n-1)   ^2,) 


«l.l(-l)""' 


■«1,2(-1)" 


<*2, 1>   <^2,3>    •••>   <*2, 


-f  ... 


+  ai 


That  is  (Art.  616), 


^*2,l'    '^'2,25    •-•)   (^2,1 


^n,  1>  tt„_2>   •••>    tt„  n_i 


=  0. 


(2) 


DETERMINANTS. 


449 


Equation  (2)  expresses  the  relation  which  must  hold 
between  the  coefficients  of  the  unknown  quantities  in  the 
given  equations  in  order  that  all  the  equations  may  be  satis- 
fied by  the  same  sets  of  values  of  the  unknown  quantities. 


EXAMPLES. 
625.   Solve  the  following  by  determinants 


i  2x  +  3y-\-    z=   4. 

1.  <     x-\-2y-\-2z=    6. 

{5x+    y  +  4.z  =  21. 

r  4a; +  5^-72;  =  -8. 

2.  J  3a;-42/-2z=   25. 

[    X  +  32/+    2  =  -9.. 


4. 


2x  +  5y-^z=      17. 

Qx  —  2y  —  5z=—   3. 

[  3a;  +  7y-f  4z  =  -18. 

2^4-  y+  z+  u=  0. 

x-2y+3z-4:u=29. 

2x+Sy-4:Z-5u=    9. 

l3x-4:y-\-5z  +  6u=    1. 


5.   Express 

3,  -2 

-8,       7    ^ 

it       6    --^« 

6.   Express 

2,  3,  -5 
1,      0,      2 

3,  -2,  -4 

X 

5,      0,  -2 

0,  -3,  -4 

-6,       1,      3 

as  a  determinant. 

6,       1,  2 

7.  Express    —4,       3,  0 

0,  -7,  5 

(Compare  Art.  622.) 

8.  Express  the  square  of 


0, 

c, 

b 

c, 

0, 

a 

b, 

a, 

0 

as  a  determinant. 


as  a  determinant. 


450  COLLEGE   ALGEBRA. 


XLI.    THEORY  OF  EQUATIONS. 

626.  Every  equation  of  the  wth  degree  (Art.  179)  involv- 
ing one  unknown  quantity,  can  be  written  in  the  form 

cc"  +  PiX"~'^  +  p.^x""^  +  •••  -Fi>„_iX+^„  =  0;  (1) 

where  the  coefficients  p^,  p2,  •■-,  Pn  may  be  positive  or  nega- 
tive, integral  or  fractional,  rational  or  irrational,  real  or 
imaginary,  or  zero. 

If  none  of  the  coefficients  2h,  P2,  ••)  Pn  are  zero,  the  equa- 
tion is  said  to  be  Complete;  if  one  or  more  of  them  are  zero, 
it  is  said  to  be  Incomplete. 

We  shall  hereafter  speak  of  (1)  as  the  General  Form  of 
the  equation  of  the  71th  degree. 

627.  It  will  be  proved  in  Appendix  II.,  that  every  equation 
of  the  above  form  has  at  least  one  root,  real  or  imaginary. 

628.  A  function  of  x  (Art.  213)  is  often  represented  by 
the  symbol /(cc),  or  F{x). 

If,  in  any  investigation,  a  certain  function  of  x  is  repre-- 
sented  by /(a;),  then,  whatever  a  may  be,  f{a)  is  taken  to 
represent  the  result  obtained  by  substituting  in  the  given 
function  a  in  place  of  x. 

Thus,  if  f{x)  =  o?  +  ^x-2,  then 

/(3)  =  32  +  3-3-2  =  9-f-9-2  =  16; 
/(-3)  =  (-3)-4-3(-3)-2  =  9-9-2=-2;  etc. 

629.  If  a  is  a  root  of  the  equation 

X"  +pia;"~^  -I +  2)„-iX  -\- Pn  =  0, 

then  the  first  member  is  divisible  by  x  —  a. 

The  division  of  the  first  member  hj  x  —  a  may  be  carried 
out  until  a  remainder  is  obtained  which  does  not  contain  x. 

Let  Q  denote  the  quotient,  and  R  the  remainder. 


THEORY   OF   EQUATIONS.  451 

Then  the  given  equation  may  be  made  to  take  the  form 
{x-a)Q  +  E  =  0.  (1) 

Since  a  is  a  root  of  the  given  equation,  equation  (1)  must 
be  satisfied  when  x  is  put  equal  to  a. 

Putting  x  =  a,we  have,  since  M  does  not  contain  x,  B  =  0. 

Therefore  aj  —  a  is  a  factor  of  the  first  member  of  the  given 
equation,  for  it  is  contained  in  it  without  a  remainder. 

630.  Conversely,  if  the  first  member  of 

ic"  -fpiX"-^  -I \-2^n~i^  -\-Pn  =  0 

is  divisible  by  x  —  a,  then  a  is  a  root  of  the  equation. 

For  since  the  first  member  of  the  given  equation  is  divisi- 
ble by  cc  —  a,  the  equation  may  be  made  to  take  the  form 

(x-a)Q  =  0; 
and  it  follows  from  Art.  350  that  a  is  a  root  of  this  equation, 

631.  It  follows  from  Art.  630  that  if  the  first  member  of 

PqX"  +  piX"~^ -\ +P„_ia;+p„  =  0 

is  divisible  by  ax  +  b,  then is  a  root  of  the  equation. 

a 

EXAMPLES. 

632.  Prove  by  the  method  of  Arts.  630  or  631 : 

1.  That  5  is  a  root  of  a;^- 2^2 -19a; +  20  =  0. 

2.  That -3isarootof  2ar'  +  3a;2_2a;  +  21  =  0. 

3.  That  -  is  a  root  of  33;"  -  8 a;^  +  13 a;-  -  9 a;  +  2  =  0. 

o 

4.  That  -  4  is  not  a  root  of  a;"  -  a;^  +  7  x  -  12  =  0. 

5.  That  -  -  is  a  root  of  8  a;'  +  6  a;"  - 15  ar  -  16  a;  -3  =  0. 

4 

6.  That  -  is  nit  a  root  of  15  ar^  +  a;^  -f-  14  a;  -  3  =  0. 


452  COLLEGE   ALGEBRA. 

633.  Number  of  Roots. 

All  equation,  of  the  nth  degree  cannot  have  more  than  n  dif- 
ferent roots. 

Let  tne  equation  be 

X"  +pia;""^  -t-2hx"-^"  -\ -i-Pn-i^+Pn  =  0;  (1) 

By  Art,  627,  thia  equation  must  have  at  least  one  root. 

Let  a  be  this  root ;  then  by  Art.  629,  the  first  member  is 
divisible  by  cc  —  a,  and  the  equation  may  be  put  in  the  form 

(X  —  a)  (X"-^  +^23?""^  -I h  Qn-l^  +  Qn)  =  0. 

Then  by  Art.  350,  the  equation  may  be  solved  by  placing 
X  —  a  =  0, 

and  «"-^  +  (h  x"^^  H 1-  g«-i  a;  +  ^„  =  0.  (2) 

Equation  (2)  must  also  have  at  least  one  root. 
Let  b  be  this  root;  then  (2)  may  be  written 

(cc  -  6)  (a;"-2  +  rgcc^s  -\ h  r„_i  a;  +  r„)  =  0, 

and  the  equation  may  be  solved  by  placing 

x-b  =  0, 
and  ic""^  +  Vgcc"^^  +  •  •  •  +  ''n-i  x  +  r„  =  0. 

Continuing  the  process  until  n  —  1  binomial  factors  have 
been  divided  out,  we  shall  arrive  finally  at  an  equation  of 
the  first  degree, 

X  —  k  =  0  ;  whence,  x  =  7c. 

Therefore  the  given  equation  has  the  n  roots  a,  b,  ...,  k. 

Note.  It  should  be  observed  that  the  roots  are  not  necessarily 
unequal ;  thus,  the  equation  a;^  —  3  x'^  +  4  =  0  can  be  written  in  the 
form  {x  +  1)  (x  —  2)  (x  —  2)  =  0,  and  its  three  roots  are  —  1,  2,  and  2. 

634.  It  is  customary  to  enunciate  the  principle  demon- 
strated in  Art.  633  in  the  following  form  : 

An  equation  of  the  nth  degree  has  n  roots; 
by  which  we  mean  that  it  may  have  n,  different  roots,  but 
cannot  have  more  than  n. 


THEORY   OF   EQUATIONS.  453 

635.  It  follows  from  Art.  633  that  if  m  roots  of  an  equa- 
tion of  the  7ith  degree  are  known,  the  equation  may  be 
depressed  to  another  of  the  {n  —  m)th  degree  which  shall 
contain  the  remaining  roots. 

Hence,  if  all  the  roots  of  an  equation  are  known  except 
two,  these  two  may  be  obtained  from  the  depressed  equation 
by  the  rules  for  quadratics. 

1.  Two  roots  of  the  equation  9  a;'' -37  o.-^- 8  a;  +  20  =  0 

are  2  and  —  -  ;  what  are  the  others  ? 
3 

Dividing  9a;V37a;2  -  8a;  +  20   by    (x- 2)(3a;  + 5),   or 

3  a;^  —  a;  —  10,  the  depressed  equation  is  3  a;-  +  a;  —  2  =  0. 

2 
Solving  by  the  rules  for  quadratics,  we  have  a;  =  -  or  —1. 

o 

EXAMPLES. 

2.  One  root  of  af'  — 37  a; +  84 —  0  is  3;  what  ?re  the 
others  ? 

3.  One  root  of  2a;3_^5a^.2_43^_  90  ^  0  is  -  2  ;  what 
are  the  others  ? 

4  One  root  of  24  x^  -  46 .«-  +  29  a;  —  6  =  0  is  - ;  what  are 
the  others  ?  ^ 

5.  One  root  of  32a;3  _  32^52  _  94^;  +  39  =  0  is  -  ^ ;  what 
are  the  others  ? 

6.  Two  roots  of  6  a;*  -  af'  -  42  a;^  +  15 a;  +  50  =  0  are  2  and 
—  1 ;  wiiat  are  the  others  ? 

7.  Two  roots  of  36a;'' -  445a;2  + 49  =  0  are  I  and  -^; 
what  are  the  others  ? 

8.  Two  ro<5ts  of  a;*  -  10  aar' +  35  aV  -  50  a^a;  +  24 a'' =  0 
are  a  and  3  a ;  what  are  the  others  ? 

9.  One  root  of  the  equation 

a:?  —  (m  +  2)x^  —  (m^+  4m  +  5)a;  +  m^  +  6 ?7i-  + 11  w  -f-  6  =  9 
is  m  -\-l;  what  arc  the  others  ? 


454  COLLEGE   ALGEBRA. 

636.   Formation  of  Equations. 

It  follows  from  Art.  633  that  if  the  roots  of 
ic"  -\-piX"-^^  +  •••  +2^n-i^  +2K  =  ^ 
are  a,  h,  ...,  k,  the  equation  may  be  written  in  the  form 
{x-a){x-h)--{x-k)  =  Q. 

Hence,  to  form  an  equation  which  shall  have  any  required 
roots, 

Subtract  each  of  the  roots  from  x,  and  jJlace  the  product  of 
the  resulting  expressions  equal  to  zero.      (Compare  Art,  354.) 

1.  Form  the  equation  whose  roots  shall  be  1,  —  6,  -  and 
_5  ^ 

4" 
By  the  rule,     {x  -  1)  (x  +  6)  fx  -  ^\  fx  +  ^\  =  0. 

Multiplying  the  terms  of  the  third  and  fourth  factors  by 
2  and  4,  respectively,  we  have 

(x  -l)(x+6){2x-  1)  (4x  +  5)  =  0. 
That  is,  8a;^  +  46ar^  -  23 ar  -  61  x  +  30  =  0. 


EXAMPLES. 

Form  the  equations  whose  roots  shall  be  : 

2.    2,  3,  5. 

6.    -2,  -2,  1|,1|. 

3.  -  2,  -  3,  -  4,  9. 

4.  1,  -  2,  3,  0. 

7     4         r,        11 

g     .        1        2       3 
^-    ^'-2'-3'~4' 

6.   6,-l,H,-|. 

9.   2±V3,  -,2±V3. 

10.  i±-^^, 

2 

-2±V5 

2 

jj     V-i±V-3  -V-i±V-3 

THEORY   OF  EQUATIONS.  455 

637.   Composition  of  Coefficients. 

By  Art.  636,  the  equation  of  the  nth  degree  whose  roots 
are  a,  b,  c,  d,  ...,  k,  I,  m,  is 

{x  -  a)  {x  -  b)  {x  -  c)  {x  -  d)  ■■■  (x  -  m)  =  0.  (1) 

By  actual  multiplication,  we  obtain 
(x  —  a)  (x—b)=x^—  (a  +  b)x-{-ab; 
(x  —  a){x—b)  {x  —  c)  =  x^—{a-\-b-^c)x^-\-{ab-\-bc-{-ca)x  —  abc; 
etc. 

When  all  the  factors  of  the  first  member  of  (1)  have  been 
multiplied  together,  we  shall  have  a  result  of  the  form 

cc"  H-Pix"-!  +  2hx"~^  +^3^""^  -\ \-Pn  =  0 ; 

where    p^  =  —  (a  +  6  +  c  +  •  •  •  +  A:  +  Z  +  ?)i) ; 
p,^  —  ab  -\-  ac  -^  be  -\-  •••  +  Im  ; 
Ps  =  —  {abc  +  abd  +  acd  +  •  •  •  +  klm)  ; 


p„=  ±  abed '••klm,  according  as  n  is  even  or  odd. 

Hence,  if  an  equation  of  the  nth  degree  is  in  the  general 
form. 

The  coefficient  of  the  second  term  is  equal  to  minus  the  sum 
of  all  the  roots. 

The  coefficient  of  the  third  term  is  equcd  to  the  sum  of  their 
products,  taken  two  at  a  time. 

The  coefficient  of  the  fourth  term  is  equcd  to  minus  the  sum 
of  their  jyi'oducts,  taken  three  at  a  time;  etc. 

The  last  term  is  equal  to  plus  or  minus  the  product  of  cdl  the 
roots,  accord'ng  as  n  is  even  or  odd. 

638.  It  follows  from  Art.  637  that,  if  an  equation  of  the 
nth.  degree  is  in  the  general  form, 

If  the  second  term  is  wanting,  the  sum  of  the  roots  is  0. 

If  the  last  term  is  wanting,  at  least  one  root  is  0. 

If  the  last  term  is  integral,  it  is  divisible  by  every  inte- 
gral root. 


456  COLLEGE   ALGEBRA. 

639.  If  all  but  one  of  the  roots  of  an  equation  of  the  «th 
degree  in  the  general  form  are  known,  the  remaining  root 
may  be  found  either  by  adding  the  sum  of  the  known  roots 
to  the  coefficient  of  the  second  term  of  the  given  equation 
and  changing  the  sign  of  the  result,  or  by  dividing  the  last 
term  of  the  given  equation  by  plus  or  minus  the  product  of 
the  known  roots  according  as  n  is  even  or  odd. 

If  all  but  two  are  known,  the  coefficient  of  the  second 
term  of  the  depressed  equation  (Art.  635)  may  be  found  by 
adding  the  sum  of  the  known  roots  to  the  coefficient  of  the 
second  term  of  the  given  equation;  and  the  last  term  of 
the  depressed  equation  may  be  found  by  dividing  the  last 
term  of  the  given  equation  by  plus  or  minus  the  product  of 
the  known  roots  according  as  n  is  even  or  odd. 

EXAMPLES. 

In  each  of  the  following  obtain  the  required  roots  by 
the  above  method  : 

1 .  Two  roots  of  ar^  —  4  x^  —  17  ic  +  60  =  0  are  —  4  and  5  ; 
what  is  the  other  ? 

2.  Three  roots  pf  a;*  -  45x2  -f  40  a;  +  84  =  0  ^re  2,  6,  and 
—  7  ;  what  is  the  other  ? 

3.  Four  roots  of  ar^  —  4a;*  —  5aT''  +  203:^  +  4.'»  —  16  =  0  are 
1,  —  1,  —  2,  and  4 ;  what  is  the  other  ? 

4.  Two  roots  of  a;*  +  2a;='  -  13  a;^  -  38  .t  -  24  =  0  are  -  1 
and  4 ;  what  are  the  others  ? 

5.  One  root  of  loa.-'  +  x*^  —  31  a;  +  15  =  0  is  —  - ;  what  are 
the  others  ? 

6.  Three  roots  of  a;*  -  74 ar^  -  24  a;-  +  937  a;  -  840  =  0  are 
1,-7,  and  8 ;  what  are  the  others  ? 

7.  Two  roots  of  2  a;"  -  13  x"  -  91  a;^  +  390  a;  +  216  =  0  are 

4  and :  what  arc  the  others  ? 

2' 


THEORY   OF   EQUATIONS.  457 

640.  Fractional  Roots. 

An  equation  in  the  general  form  whose  coefficients  are  inte- 
gral,  cannot  have  as  a  root  a  rational  fraction  (Art.  154)  in 
its  lowest  terms. 

Let  the  equation  be 

where  pi,  p2,  •••■>  Pn  are  integral. 

If  possible,  let  -,  a  rational  fraction  in  its  lowest  terms, 
h 
be  a  root  of  the  equation  ;  then. 

Multiplying  each  term  by  6""*,  and  transposing, 

6 

By  hypothesis,  a  and  h  have  no  common  divisor. 

We  then  have  a  rational  fraction  in  its  lowest  terms 
eqiial  to  an  integral  expression,  which  is  impossible. 

Therefore  the  equation  cannot  have  as  a  root  a  rational 
fraction  in  its  lowest  terms. 

641.  Imaginary  Roots. 

If  a  pure  imaginary  or  complex  number  (Art.  327)  is  a 
root  of  an  equation  in  the  general  form  with  real  coefficients, 
its  conjugate  (Art.  337)  is  also  a  root. 

Let  the  equation  be 

X"  +  pi  x"-^  H f-  p„_i  x-\-p„  =  0,  (1 ) 

where  p^,  ...,  p,^  are  real  numbers. 

Let  a  +  ftV—  1,  where  a  and  &  have  the  same  meanings 
as  in  Art.  327,  be  a  root  of  the  equation ;  then, 

(a  +  bV^iy  +Pi(a  -f  6V^=l)"-i  +  ... 

+  P„  i(a  +  ^V-^)  -i-Pn  =  0. 


458  COLLEGE   ALGEBRA. 

Expanding  by  the  Binomial  Theorem,  we  have  by  Art.  333, 

»     17       / ^  n(n  —  1)      „_o,o 

n(w-l)(/i-2)       3,3   / — ^        ^^ 

+  Pi[a''-^+(n-l)a"-^6V^-  (^'~^H^^-2)  ^,.-3^2 -] 

l£ 

+  -  +P„-i  (a  +  W-  1)  +1\  =  0.  (2) 

Collecting  the  real  and  imaginary  terras,  we  shall  have  a 
result  of  the  form 

where  P  and  Q  are  real. 

In  order  that  this  equation  may  hold,  we  must  have 

P  =  0,  and  Q  =  0. 

Now  substituting  a  — &V— 1  for  x  in  the  first  nxember 
of  (1),  it  becomes 

(a  -  6 V^l)"  -\-lh{ci  -  6 V^T)"-^  +-' 

+  P«-i(a-&V^^)+i\.  (3) 

Expanding  the  powers  of  a  — 6V— 1,  we  shall  have  a 
result  which  differs  from  the  first  member  of  (2)  only  in 
having  the  odd  terms  in  each  expansion,  or  those  involving 
V  —  1  as  a  factor,  changed  in  sign. 

Then,  collecting  the  real  and  imaginary  terms,  the  expres- 
sion (3)  is  equal  to 

where  P  and  Q  have  the  same  meanings  as  before. 
But  since  P  =  0  and  Q  =  0,  P  -  Q  V^^  =  0. 
Therefore  a  —  6V—  1  is  a  root  of  (1). 


THEORY   OF   EQUATIONS.  459 

Note.  The  product  of  the  factors  of  the  first  member  of  (1),  Art. 
641,  coiTesponding  to  the  conjugate  imaginary  roots  a+  6\/^^  and 
a  —  bV—  1,  is 

[X  -  (a  +  i)  V^n:)]  [x  -  (a  -  &  V^ ] 

=  (a;  -  a)2  -  (6  V^)- =  (x  -  «)2  +  62  ; 

and  is  therefore  positive  for  every  real  value  of  x. 

642.  It  follows  from  Arts.  G33  and  641  that  every  equa- 
tion of  odd  degree  has  at  least  one  real  root ;  for  an  equation 
cannot  have  an  odd  number  of  imaginary  roots. 


TRANSFORMATION   OF  EQUATIONS. 

643.   To  transform  an  equation  into  another  which  shall 
have  the  same  roots  loith  contrary  signs. 
Let  the  equation  be 

a;"  +  xh  •'^"^  +  Ih  a^*"~^  H h  l>«-i  x+p,^  =  0.  ( 1 ) 

Substituting  —  y  for  x,  we  have 

{-yy+Pi{-yY-^+ih{-yY''-\-  •••  +i^«-i(-2/) +p„=  o. 

That  is, 

or,  2/"  -  Pi  2/""^  +  P-'.V""^ ±  Pn-i  yTPn  =  Q;  (2) 

the  upper  or  lower  signs  being  taken  according  as  n  is  odd 
or  even. 

It  follows  from  (1)  and  (2)  that  the  desired  transforma- 
tion may  be  effected  by  simply  changing  the  signs  of  the 
alternate  terms  beginning  with  the  second. 

Note.  If  the  equation  is  incomplete,  any  missing  term  must  be 
supplied  with  the  coefficient  zero  before  applying  the  rule. 


460  COLLEGE   ALGEBRA. 

Example.  Transform  the  equation  r»^  —  10  ic  +  4  =  0  into 
another  which  shall  have  the  same  roots  with  contrary 
signs. 

The  equation  may  be  written 

a;3  +  0-a;--10x  +  4=0. 
Then  by  the  rule,  the  transformed  equation  is 

jc3_0.a^-10a;-4  =  0,  or  .t'' -  lOx- 4  =  0. 

-,    644.    To  transform  an  equation  into  another  whose  roots 
shall  he  m  times  those  of  the  first. 
Let  the  equation  be 

x^  +j)ix""^  +^2-'^""^  +  •••  +  Pn~\^  +Pn  =  ^- 

Substituting  ^  for  x,  whence  y  =  mx,  we  have 
m 

(i:)"+^<i)"-+-(0-+-+-.(i')+-="- 

Multiplying  each  term  by  m", 

?/"  ■i-pimy"-^  ^psm^y"-^  +  -"  +Pn-i'm"~'^y  -\r2^nin"  =  0. 

Hence,  to  effect  the  desired  transformation,  multiply  the 
second  term  by  m,  the  third  term  by  m-,  and  so  on. 

Example.  Transform  the  equation  ar'  +  7a^  —  6  =  0  into 
another  whose  roots  shall  be  4  times  those  of  the  first. 

Supplying  the  missing  term  with  the  coefficient  zero,  and 
applying  the  rule,  we  have 

ar5  +  4 .  7x-2  +  42.  Oa;  -  4=.  6  =  0,  or  a;3  4.  28rc2  -  384  =  0. 

645.  .  To  transform  an  equation  loitli  fractional  coefficients 
into  another  whose  coefficie)its  shall  be  integral,  that  ofthefir.st 
term  being  unity. 

The  transformation  may  be  effected  by  multiplying  the 
roots  of  the  equation  by  m  (Art.  644),  and  then  giving  to 
m  such  a  value  as  will  make  all  the  coefficients  integral. 


THEORY   OF   EQUATIONS.  461 

By  giving  to  m  the  least  value  which  will  make  all  the 
coefficients  integral,  the  result  will  be  obtained  in  its 
simplest  form. 

Example.  Transform  the  equation  ^^  —  V  ~  ^  +  Tnc  "^  ^^ 
into  another  whose  coefficients  shall  be  integral,  that  of  the 
first  term  being  unity. 

Multiplying  the  roots  by  m,  we  obtain 

ar ar x-\ =  0. 

3  3G        108 

It  is  evident  by  inspection  that  the  least  value  of   m 

which  will  make  all  the  coefficients  integral,  is  6 ;  putting 

m  =  6,  we  have 

x'-2x--x  +  2  =  Q, 

whose  roots  are  6  times  those  of  the  given  equation. 

646.  To  transform  an  equation  into  another  wJiose  roots 
shall  be  those  of  the  first  increased  by  m. 

Let  the  equation  be 

Substituting  y  —  m  for  x,  whence  y  =  x  +  m,  we  have 
{y  -  my  +i9i  {y  -  my~'  +  •••  +p„_i  (y  -  m)  +  p„  =  0.    (2) 

Expanding  the  powers  of  y  —  m  by  the  Binomial 
Theorem,  and  collecting  the  terms  involving  like  powers 
of  y,  we  shall  have  a  result  of  the  form 

yn  j^  ^^^ yu-i  +  . . .  +  q^^_^ y  +  q,,  =  0,  (3) 

whose  roots  are  those  of  the  given  equation  increased  by  m. 

647.  If  m  and  the  coefficients  of  the  given  equation  are 
integers,  the  coefficients  of  the  transformed  equation  may 
be  conveniently  found  by  the  following  method. 


t62  COLLEGE   ALGEBRA. 

Putting  X  +  111  for  y  in  (3),  we  obtain 
(a;  +  my  +  q,  {x  +  my-'  +  •••  +  g„_i  (a;  +m)  +  7.  =  0 ;     (4) 

which  must,  of  course,  take  the  same  form  as  (1)  on  ex- 
panding the  powers  of  x  +  m,  and  collecting  the  terms 
involving  like  powers  of  x. 

Dividing  the  first  member  of  (4)  by  cc  +  m,  we  have 

{x  +  m)"-i  +  qi{x  +  my-^  +  ---+  q„_._  {x  +  m)  +  g„_i     (5) 

as  a  quotient,  with  a  remainder  g„. 

Dividing  (5)  by  x  +  m,  we  have  the  remainder  q,^_-^ ;  etc. 

Hence,  to  obtain  the  coefficients  of  the  transformed 
equation : 

Divide  the  first  member  of  the  given  equation  hyx-\-m; 
the  remainder  will  be  the  last  term  of  the  reqtiired  equation. 

Divide  the  quotient  just  found  by  x  +  m ;  the  remainder 
will  be  the  coefficient  of  the  next  to  the  last  term  of  the  trans- 
formed equation;  and  so  on. 

648.  To  transform  an  equation  into  another  whose  roots 
shall  be  those  of  the  first  diminished  by  m,  we  change  y  —  m 
to  y  -i-m  in  the  method  of  Art.  646,  and  x-\-mtox  —  m  in 
the  rule  of  Art.  647. 

649.  Transform  the  equation  .r''  —  7  a^  +  6  =  0  into  another 
whose  roots  shall  be  those  of  the  first  increased  by  2. 

We  may  either  substitute  y  —  2  for  x  in  the  given  equa- 
tion, or  use  the  rule  of  Art.  647. 

In  the  latter  case,  dividing  x^  —  7  a;  -f  6  by  .t  +  2,  we  have 
a;-  —  2x  —  3  as  a  quotient,  and  12  as  a  remainder. 

Dividing  .r^  — 2.^  —  3  by  x-\-2,  we  have  x  — 4  as  a 
quotient,  and  5  as  a  remainder. 

Dividing  a;  —  4  by  a;  +  2,  we  have  the  remainder  —  6. 

Then  the  transformed  equation  is 

c^-6x''  +  5x  +  12  =  0. 


THEORY   OF   EQUATIONS.  463 

650.   Synthetic  Division. 

The  operation  of  division,  in  examples  like  that  of  Art. 
64:1),  may  be  conveniently  performed  by  a  process  known 
as  Synthetic  Division. 

Let  it  be  required  to  divide  ar^  —  12  a;- +  29  a;  — 21  by 
a;  —  3. 

Using  detached  coefficients  (Art.  105),  we  have 


1  -  12  +  29  - 

-21 

1-3 

1-    3 

1-9  +  2,  Quotient. 

-    9 

-    9  +  27 

+    2 

+    2- 

-   6 

- 

-15, 

Eemainder. 

We  may  omit  the  first  term  of  each  partial  product,  for 
it  is  merely  a  repetition  of  the  term  immediately  above. 

Also,  the  second  term  of  each  partial  product  may  be 
added  to  the  corresponding  term  of  the  dividend,  provided 
we  change  the  sign  of  the  second  term  of  the  divisor  before 
multiplying. 

The  work  now  stands  : 

1-12  +  29-21   I  1+3 
+   3  I  1-9  +  2 

-   9 

-27 

+    2 

+    6 

-15 

The  first  term  of  the  divisor  being  unity  in  all  applica- 
tions of  Art.  647,  it  may  be  omitted ;  and  the  first  terms 
of  the  successive  dividends  constitute  the  quotient. 


464  COLLEGE   ALGEBRA. 

Eaising  tlie  oblique  columns,  the  operation  will  stand  as 
follows : 

Dividend,  i   _  12  +  29    -21   [  +3 

Partial  j^roducts,        +    3—27    +    6 

Quotient,  1   —    9  +    2,-15  Remainder. 

The  complete  result  is  obtained  as  follows  : 

Multiplying  the  first  term  of  the  dividend  by  3,  and  add- 
ing the  result  to  the  second  term  of  the  dividend,  gives  the 
second  term  of  the  quotient. 

Multiplying  the  latter  by  3,  and  adding  the  result  to  the 
third  term  of  the  dividend,  gives  the  last  term  of  the 
quotient. 

Multiplying  the  latter  by  3,  and  adding  the  result  to 
the  last  term  of  the  dividend,  gives  the  remainder. 

Therefore  the  quotient  is  cc^  —  9  x  +  2,  and  the  remainder 
-15. 

Note.  If  the  term  involving  any  power  of  x  is  wanting,  it  must  be 
supplied  with  the  coefficient  0  before  applying  the  rule. 

By  the  method  of  Synthetic  Division,  the  work  of  trans- 
forming the  equation 

into  another  whose  roots  shall  be  greater  by  2  (compare 
Art.  649),  will  stand  as  follows  : 

1  +0   -7  -f    6   I  -2 

zl  ±i  ±± 

_  2  -  3  +12,  1st  Rem. 
-2  +8 

-  4  +  5,  2d  Rem. 

-  2 

-  6,  3d  Rem. 
Thus  the  transformed  equation  is 


THEORY   OF   EQUATIONS.  465 


EXAMPLES. 

651.   Transform  each  of  the  following  into  an  equation 
which  shall  have  the  same  roots  with  contrary  signs  : 

1.    x4_^3rc^-2;c2-5a;  +  7  =  0.       2.    cc^  -  5a;- +  16  =  0. 

3.  Transform  the  equation  a.-^  —  5  a;-  —  7  a;  +  11  =  0  into 
another  whose  roots  shall  be  those  of  the  first  multiplied 
by  3. 

4.  Transform  the  equation  a;^  +  6ar'  —  2a;  —  5  =  0  into 
another  whose  roots  shall  be  those  of  the  first  multiplied 
by  -5. 

5.  Transform  the  equation  2 a;'^— 5 a; +7=0  into  another 

2 
Avhose  roots  shall  be  those  of  the  first  multiplied  by  -• 

o 

6.  Transform  the  equation  6a;^  —  3ar''+  8ar^—  64  =  0  into 
another  whose  roots  shall  be  those  of  the  first  multiplied 

by-^- 

Transform  each  of  the  following  into  an  equation  with 
integral  coefficients,  that  of  the  first  term  being  unity  : 


¥+1-1= 

:0. 

^    3        27      72 

f-i=- 

10.    ^-y^-^  +  :^  =  0. 
4o        to       biO 

sform  the  eq 

ua 

tion  ar^  +  2a;-  -  7a;  -  72  =  0  into 

another  whose  roots  shall  be  less  by  4. 

12.    Transform   the    equation   ar^  —  19  a;- +  268  =  0   into 
another  whose  roots  shall  be  less  by  5. 

13     Transform  the  equation  a;'  + 5a;-  +  4a;  — 23  =  0  into 
another  whose  roots  shall  be  greater  by  1. 

14.    Transform  the  equation  a;^  — ar'  — 2a;- +  7.t  —  81  =  0 
into  another  whose  roots  shall  be  greater  by  3. 


4d6  college  algebra. 

15.  Transform  the  equation  aj^  +  3a^  — 5a;4-2  =  0  into 
another  whose  roots  shall  be  less  by  6. 

16.  Transform  the  equation  x'^  —  9af—(jx—12  =  0  into 
another  whose  roots  shall  be  greater  by  4. 

652.  To  transform  the  eqiiation 

X"  +2hX"-'^  H ^Pn-l^+Pn  =  0, 

where  p^  is  not  zero,  into  another  whose  second  term  shall  he 
wanting. 

Expanding  the  powers  of  y  —  m  in  the  first  member  of 
(2),  Art.  646,  and  collecting  the  terms  involving  like  pow- 
ers of  y,  we  have 

y"  +  ilh  -  mn)y"''^  -\ =0. 

If  m  is^'so  taken  that  p^  —mn  =  0,  whence  m  =  ~,  the 
coefficient  bf  z/"~^  will  be  zero. 

Hence  the  desired  transformation  may  be  effected  by  sub- 
stituting in  the  given  equation  y  — ^  in  place  of  x. 

:       DESCARTES'  RULE   OF  SIGNS. 

653,  Ncite  1.  If  an  equation  of  the  »itli  degree  is  in  the  general 

form  (Art.  6^6),  a  Permanence  of  sign  occiu's  when  two  successive 
terms  have  the  same  sign,  and  a  Variation  of  sign  occurs  when  two 
successive  terms  have  opposite  signs. 

Thus,  m  the  equation  x^  —  3  x*  —  a:^  +  5  x  +  1  =  0,  there  are  two  per- 
manences and  t,wo  variations. 

Descartesi<Riile.  No  equation,  whether  complete  or  incom- 
plete, can  have  a  greater  number  of  positive  roots  than  it  has 
variations  0  sign;  and  no  <^omplete  equation  can  have  a 
greater  nurrHber  of  negative  roots  than  it  has  permanences  of 
sign.  I 

Let  an  equation  in  tlie  general  form  have  the  following 
signs, 

+  +()_+()  0 -f  0  0  ()-  +  +()  + 

th(:  missing  terms  being  supplied  witli  zero  coeiiicients. 


THEORY   OF   EQUATIONS.  467 

If  we  introduce  a  new  positive  root  a,  we  multiply  this 
hj  X  —  a  (Art.  636)  ;  writing  only  the  signs  which  occur  in 
the  process,  we  have 

1      2      3      4      5      G      7      8      9     10    11    12    13    14    15    16    17    18 

+  +o--i-oo---i-ooo-4-+o+  (i) 


++o_+oo--+ooo-++o+ 

__o +-0 0 ++-0 0 0 +--0- 
+  m--  +  -0-m-{--00--i-m-  +  -  (2) 

1      2      3      4      5      6      7      8      9     10    11    12    13    14    15    16    17    18    19 

where  m  signifies  a  term  which  may  be  +,  0,  or  — . 

Now,  in  each  of  the  expressions  (1)  and  (2),  let  a  dot  be 
placed  over  the  first  minus  sign,  then  over  the  next  plus 
sign,  then  over  the  next  minus  sign,  and  so  on ;  no  account 
being  taken  of  the  terms  marked  m  in  (2). 

The  number  of  dots  shows  the  number  of  variations  in 
(1),  and  the  least  number  of  variations  in  (2) ;  thas,  in  (1) 
there  are  six  variations,  and  in  (2)  at  least  nine. 

In  the  above  result  we  observe  the  following  laws  : 

I.  If  a  dotted  term  in  (1)  follows  a  term  of  unlike  sign, 
as  5,  10,  or  15,  the  corresponding  term  of  (2)  is  dotted. 

II.  If  a  dotted  term  in  (1)  follows  one  or  more  ciphers, 
as  4,  8,  or  14,  that  term  of  (2)  under  the  left-hand  cipher 
is  dotted ;  as  3,  6,  or  11. 

III.  The  last  term  of  (2)  is  dotted. 

It  follows  from  the  above  that  the  introduction  of  a  new 
positive  root  increases  the  number  of  variations  in  the 
equation  by  at  least  one ;  for  to  each  dot  in  (1)  there  corre- 
sponds one  in  (2),  and  besides  the  last  term  of  (2)  is  dotted. 

If,  then,  we  form  the  product  of  ail  the  factors  corre- 
sponding to  the  negative  and  imaginary  roots  of  an  equa- 
tion, multiplying  the  result  by  the  factor  corresponding  to 
each  positive  root  introduces  at  least  one  variation. 


468  COLLEGE   ALGEBRA. 

Hence  the  equation  cannot  have  a  greater  number  of 
positive  roots  than  it  has  variations  of  sign. 

Note  2.  That  the  above  laws  hold  universally  may  be  demonstrated 
as  follows ;  m  denoting,  as  before,  a  term  which  may  be  either  +,  0, 
or  — : 

I.  Since  the  signs  of  all  the  terms  of  an  equation  may  be  changed 
without  altering  the  values  of  its  roots,  the  only  cases  which  can  occur 
are:  (a.),  a  —  sign,  and  then  r  successive  +  signs,  followed  by  a 
dotted  —  sign  ;  (&.),  a  cipher,  and  then  r  successive  +  signs,  followed 
by  a  dotted  —  sign. 

(«•)       _  (6.) 

-+  +  ••■  +  -  0  +  +  --  +  - 


+  - 

-  +  +  • 
+  -• 

■■  +  — 
+ 

0  +  +  --  +  - 
0 + 


■\-m---m — 1-  +m---m — \- 

n.  The  only  cases  which  can  occur  are:  (c),  a  —  sign,  then  r 
successive  +  signs,  and  then  q  successive  ciphers,  followed  by  a  dotted 
—  sign;  (ii.),  a  cipher,  then  r  successive  +  signs,  and  then  q  suc- 
cessive ciphers,  followed  by  a  dotted  —  sign. 

(c.)        _  {d.) 

—  +  +  —  +  0  0  •••  0  -        0  +  +  •••  +  0  0  •••  0  — 

+  -  +- 


...+  00---0-        0  +  +  •••+  0  0  •••  0  - 
0  ...  0  0  +        0 0  ...  0  0 + 


+  7n...?)t— O-.-O— +         jf.m---m—  a  ••■  a  —  + 

III.  The  above  results  demonstrate  the  third  law  in  all  cases 
except:  (e.),  a  —  sign,  and  then  r  successive  +  signs  at  the  end  of 
the  equation  ;  (/.),  a  cipher,  and  then  r  successive  +  signs  at  the  end 
of  the  equation, 

-  +  +  •••+.  0 +  +  ...+ 

+  -  +- 


-  +  +  •••+  0+  + 

+ 0- 


THEORY   OF   EQUATIONS.  469 

To  prove  the  second  statement  in  Descartes'  Rule,  let  —y 
be  substituted  for  x  in  any  complete  equation;  then  since 
the  signs  of  the  alternate  terms  beginning  with  the  second 
are  changed  (Art.  643),  the  original  p'ermanences  of  sign 
become  variations. 

But  the  transformed  equation  cannot  have  a  greater  num- 
ber of  positive  roots  than  it  has  variations;  and  hence  the 
original  equation  cannot  have  a  greater  number  of  negative 
roots  than  it  has  permanences. 

Note  3.  In  all  applications  of  Descartes'  Rule,  the  equation  must 
contain  a  term  independent  of  x ;  that  is,  no,  root  must  be  equal  to 
zero  (Art.  350)  ;  for  a  zero  root  cannot  be  considered  as  either  positive 
or  negative. 

654.  It  follows  from  the  last  part  of  Art.  653  that  in  any 
equation,  whether  complete  or  incomplete,  the  number  of 
negative  roots  cannot  exceed  the  number  of  variations  in 
the  equation  which  is  formed  from  the  given  equation  by 
changing  the  signs  of  the  alternate  terms  beginning  with 
the  second. 

655.  In  any  compilete  equation,  the  sum  of  the  number  of 
permanences  and  variations  is  equal  to  the  number  of  terms 
less  one,  or  to  the  degree  of  the  equation  ;  that  is,  to  the 
number  of  roots  (Art.  633). 

Hence,  if  the  roots  of  a  complete  equation  are  all  real, 
the  number  of  positive  roots  is  equal  to  the  number  of  vari- 
ations, and  the  number  of  negative  roots  is  equal  to  the 
number  of  permanences. 

An  equation  whose  terms  are  all  positive  can  have  no 
positive  root ;  and  a  complete  equation  whose  terms  are 
alternately  positive  and  negative  can  have  no  negative  root. 

656.  1 .  Determine  the  nature  of  the  roots  of 

There  is  no  variation,  and  consequently  no  positive  root. 


470  COLLEGE   ALGEBRA. 

Changing  the  signs  of  the  alternate  terms  beginning  with 
the  second  (Art.  654),  we  have  x'^  +  2aj  —  5  =  0. 

(Compare  Note,  Art.  643.) 

In  this  there  afo  two  variationL  and  therefore  the  given 
eqaation  cannot  have  more  than  i>ffrr  negative  root^. 

Hence,  since  the  equation  has  three  roots  (Art.  633),  one 
of  them  must  be  negative  and  the  other  two  imaginary 
(Art.  642). 

Note.  If  two  or  more  successive  terms  of  an  equation  are  wanting, 
it  follows  by  Descartes'  Rule  that  the  equation  must  have  imaginary 
roots. 

EXAMPLES. 

The  roots  of  the  following  equations  being  all  real,  de- 
termine their  signs  : 

2.  2a;^-3a.-2- 17a; +  30  =  0. 

3.  3ar^- liar- 19a; -5  =  0. 

4.  a;''-8ar^  +  17a.-2  +  ^a;-24  =  0. 

5.  a;"  -  53 ar'  + 441  =  0. 

6.  4a;^  +  28a;'^  +  39ar'-7a;-10  =  0. 

7.  ar'-41ar''  +  12a;2  +  292a;  +  240  =  0. 

8.  3a.-5  -  2x^  -  45ar'  +  92a;  -  48  =  0. 

Determine  the  nature  of  the  roots  of  the  following : 
9.    a;3_8a;2-12  =  0.    12.    a;^  -  4x^- 5  =  0. 

10.  a;*  +  3ar  +  l  =  0.      13.    a;-' +  2x'' +  3a;2  + 1  =  0. 

11.  a;^  +  l  =  0.  14.    a;''  +  2;c--l  =  0.    (Put  ar  =  ?/.) 

DIFFERENTIATION. 

657.  We  shall  first  demonstrate  three  propositions  in 
rocrard  to  limits. 


THEORY   OF   EQUATIONS.  471 

I.  The  limit  of  the  sum  of  any  number  of  variables  is  the 
sum  of  their  Hunts. 

Let  x',  y\  z',  ...,  be  the  limits  of  the  variables  x,  y,  z,  ...  . 

Then  x'—x,  y'—y,  z'—z,  ...,  are  variables  which  can  be 
made  less  than  any  assigned  quantity,  however  small  (Art. 
209,  Note). 

Therefore,       {x'-  x)  +  {y'-  y)  +  (z'- z')+ ■■■ , 
or  {x'+y'+z'+---)-{x  +  y+z+---), 

can   be   made   less    than   any   assigned  quantity,  however 
small. 

Hence,  x'+y'+z'-\ is  the  limit  of  x  +  y  +  z-\ 

II.  The  limit  of  the  product  of  a  constant  and  a  variable  is 
the  constant  midtiplied  by  the  limit  of  the  variable. 

Let  a  be  a  constant,  and  let  x'  and  x  have  the  same  mean- 
ings as  above. 

Then  a{x'—x),  or  ax'— ax,  can  be  made  less  than  any 
assigned  quantity,  however  small. 

Whence,  ax'  is  the  limit  of  ax. 

III.  The  limit  of  the  product  of  any  number  of  variables 
is  the  product  of  their  limits. 

Let  x',  y',  z',  ...,  and  x,  y,  z,  ...,  have  the  same  meanings 
as  above,  and  let  x'—x=  I,  y'  —  y  =  7n,  z'—z  =  n,  .... 

Then  I,  m,  n,  ...,  are  variables  which  can  be  made  less 
than  any  assigned  quantity,  however  small. 

Now,  x'y'z'---  =  {x  -h  I)  (y  +  m){z  +  n)--- 

=  xyz---  +  terms  involving  I,  m,  n,  ...  . 
Whence, 
x'y'z' '••  -xyz---  =  terms  involving  I,  m,  n,  ...  .  (1) 

The  second  member  of  (1)  can  be  made  less  than  any 
assigned  quantity,  however  small. 

Therefore,  x'y'z' ...  is  the  limit  of  xyz... . 


472  COLLEGE   ALGEBRA. 

658.  Derivatives. 

Ill  any  function  of  x  (Art.  213),  let  x  +  7i  be  substituted 
for  x;  subtract  from  the  result  the  given  function,  and 
divide  the  remainder  by  h. 

The  limiting  value  of  the  result  as  h  approaches  the  limit 
zero,  is  called  the  derivative  of  the  function  luith  respect  to  x. 

Let  it  be  required,  for  example,  to  find  the  derivative  of 

with  respect  to  x. 

Substituting  x  +  h  for  x,  and  subtracting  from  the  result 
the  given  function,  we  have 

(x  +  hy-  2{x  +  hy+  5  -  (ar*  -  2ar^  +  5) 
=  3  xVi  +  3xh^  +  h^  -  4  xJi  -  2 1i\ 
Dividing  this  result  by  h,  we  have 

3x^  +  3xh  +  h''-4.x-2h.  (1) 

The  limiting  value  of  (1)  as  h  approaches  0,  is  3  x^  —  4  a;. 
Hence  the  derivative  of  ar*  —  2  x^  +  5  with  respect  to  x  is 
3  ar^  —  4  a;. 
.  The  process  exemplified  above  is  called  Differentiation. 

659.  In  general,  let  u  represent  any  function  of  x ;  and 
suppose  that,  when  x  is  changed  to  x  +  h,  u  becomes  u  -\-  h'. 

Then  the  derivative  of  u  with  respect  to  a;  is 


lim 
h  =  0 


nu  +  h')-u 


where    ^™    is  used  as  an  abbreviation  for  "  the  limit  as  h 
h  =  0 

approaches  zero  of." 

It  follows  from  the  above  that 

d  lira  r/i'l  /-i\ 

where  —u  stands  for  the  derivative  of  u  with  respect  to  x. 
dx 


THEORY   OF   EQUATIONS  473 

660.  The  process  of  (iifferentiation  is  much  facilitated 
by  means  of  the  following  formulae,  in  which  a  and  n  repre- 
sent constants,  and  u,  v,  w,  ...,  any  functions  of  x: 

I.    ^x  =  l. 
dx 

TT  f^     /  .  \  d 

II.    — (u-\-a)  =  — u. 

dx  '  dx 

TTT      d  ,     .  d 

III.  — (au)  =  a — u. 
dx  dx 

TTT      d  ,      ,       ,        ,        .        d       ,    d      ,    d 

IV.  —  {a  +  v  +  w  -\-  •••)  =  —  w-l v-\ w  -\-  ■••  • 

dx  dx        dx        dx 

V.    —  iuvvj  ■•■)  =  (vw  •  •  • )  -^ w  -f-  (uw  •••)  —  V  -\ • 

dx  dx  dx 

VI.     —  (U")  =  71U"-    ^ M. 

dx  dx 

VII.    —  (aa;")  =  wax"~^ 
dx 

661.  In  proving  the  formulae  of  Art.  660,  we  shall  sup- 
pose that,  when  x-\-h  is  substituted  for  x,  u,  v,  w,  ...,  are 
changed  in  consequence  to  m  -f  h',  v  -f-  h",  w  +  h'",  — 

Proof  of  I.     By  Art.  658, 

f^  o.  -    lini   [{x  +  li)  —  x'^  _  ^ 
dx         h  =  (^l  h  \~ 

e  derivative  with  respect  to  : 

[I. 

—  (u  +  a)=   ^™  \{u-lr^l'-[-a)-{u  +  a)'^  ^   lim  f/i^l  ^  ^  ^^ 
dx^  '       /i  =  0|_  /i  J      /t  =  o|_/iJ      dx    ' 

by  Art.  6.59,  (1). 

That  is,  the  derivative  with  respect  to  x  of  a  function  of  x 
plus  a  constant  is  equal  to  the.  derivative  of  the  function  of  x. 

For  example,  -^  {3x- ~5)=  —  (?^x"). 
dx  dx 


That  is,  the  derivative  with  respect  to  x  of  x  itself  is  unity. 
Proof  of  II. 


474  COLLEGE   ALGEBRA. 

Proof  of  III. 
d   ,      .         lim   To  («  + /i')  —  awl         lim  VaV'~\ 

d^^'"^=/.^oL — I — j=/.^o[tJ 

=  a  X  ^^'|!^'_  T-lcArt.  657,  II.)  =  a  —  u  (Art.  659,  (1)  ). 

That  is,  tlie  derivative  ivith  resjiect  to  x  of  a  constant  times 
a  function  of  x  is  equal  to  the  constant  times  the  derivative  of 
the  function  of  x. 

For  example,  —  (3x-)  =  3—  (x'). 
dx  dx 

Proof  of  lY. 

—  (?i  +  v  +  z«H ) 

dx 

^    lim   r{u  +  h'  +  v  +  h"  +  iv  +  h"'  +  ..-)-{u  +  v  +  io  +  '--)l 

h=o\_  h  J 

im   rh'+h"+h"'+...l 

RH   ,     lim   p"1   ,     lim   R"n,  /A,.t 


h 
lim 


657,  I.) 


d       ,    d      ,    d 

=  — u-\ v-\ w  + 

dx        dx        dx 


That  is,  the  derivative  with  respect  to  x  of  the  sum  of 
any  number  of  functions  of  x  is  equal  to  the  sum  of  their 
derivatives. 

Proof  of  V.     Consider  first  the  case  of  two  factors. 

(I  („„^  _    lim   r{u  +  h'){v  +  h")-uv'] 
dx  ^''"^  -h  =  0 1  I  J 

^    lim   ruh"+{v  +  h")h'~\ 
-      h  =  o\_  h  J 

„^    lim   [/*""].     lim   r,   ,    /jH^     lira   RH 
by  Art.  657,  I.  and  III. 


THEORY   OF   EQUATIONS.  475 

As  h  approaches  the  limit  0,  h"  also  approaches  0,  and 
therefore  the  limiting  value  of  v  ■\-  7i"  is  v. 

Whence,  —  (uv)=  u —v +v  —  u.  (1) 

clx  dx  clx 

Consider  next  the  case  of  three  factors. 

—  hivio)  =  -  r  (uv)  ■  ?«]  =  IV  ^^  (uv)  +110-^  10,  by  (1) 
dx  dx"-  dx  dx 


d 

-f-  uv  —  w 
dx 


f    d      ,       d     \ 
\   dx  dx    J 

d       .  d       ,         d 

=  vw  —  u-\-  uio  —  V  +  uv  -—  w. 
dx  dx  dx 

In  like  manner  the  theorem  may  be  proved  for  any  num- 
ber of  factors. 

That  is,  the  derivative  with  respect  to  x  of  the  product  of  any 
number  of  functions  of  x  is  equal  to  the  sum  of  the  results 
obtained  by  multix)lying  the  derivative  of  each  factor  by  all  the 
other  factors. 

For  example,  —[(•x  +  l)^-]  =  (.x-  +  l)-^(arO+ar^-^(aJ+l). 
dx  dx  ax 

Proof  of  YI. 

d  /...x        lim   [{n  +  h'Y-u-'\ 
dx^''^--h^^\_  h  J 

Expanding  by  the  Binomial  Theorem,  and  cancelling  %", 

r         n     17/    ,     n(n—l)       „_2  7f2     I  1 

nu"-^h'-\ 5^-^^ — Lu"-  ^h'^+  ••• 

d  ,    „x         lim  i- 

d^.(^)  =  .^o   h 


Vnn    pl!"!^    lim   [,,,,.-!+ !L(!iZ^ ,,-2;,,+ 


lim  nn    lim  r^^,^,. 

^Olh]     h=0\_ 
by  Art.  G57,  III 


476  COLLEGE   ALGEBRA. 

As  ^  approaches  the  limit  0,  h'  also  approaches  0,  and  the 
limiting  value  of  the  second  expression  in  brackets  is  nit""\- 

Whence,  —  (li")  =  nu'^~^  —  ii. 

dx  dx 

For  example,  ~\_{x^  +  1)^]  =  3  (x-  +  ly^  (x"  +  1). 
dx  dx 

Proof  of  VII. 

By  III.,  —  (ax")  =  a—  (x«)  =  anx"-'-^x,  by  VI., 
'  dx^       ^        dx^     ^  dx  '    ^        ' 

=  a?ia;"~\  by  I. 

That  is,  the  derivative  with  respect  to  x  of  a  constant  times 
any  power  of  x  is  equal  to  the  constant,  times  the  exponent  of 
the  power,  times  x  raised  to  a  power  whose  exponent  is  less 

For  example,  —  (3  x')  =  12  x;\ 
dx 


EXAMPLES. 
662.   1.    Find  the  derivative  with  respect  to  x  of 

2x^-5x^  +  7x-6. 
By  II.  and  IV.,  Art.  661, 

—  (2a;^  -  5x^  +  7a;  -6)  =  —  (2x')  -  ^-  {5x')  +  —  (7a-) 
dx^  ^      dx^       ^      dx^       '      dx^      ^ 

=  6x--10a;  +  7,  by  VII. 
Find  the  derivatives  with  respect  to  x  of  the  following : 

2.  3^2-5.  6.    6.'c''  +  ar^-12a;2  +  8x-2. 

3.  2x'-x'  +  x.  7.    5x''-2.t'-.9x-'-6x2.  • 

4.  x'  -lx''  +  2x'  +  l.        8.    x"  -\-8x'  +  10x^  -2X  +  1. 

5.  3.r^  +  5.x-''-llx\  9.    ?,x''-Sx'  +  x'-iryx-12. 


THEORY   OF   EQUATIONS.  477 

10.  Find  the  derivative  witli  respect  to  x  of  (.^•-  +  l)'l 

ByVI.,f-[(a)^+l)«]  =  3(x^+l)^|-(^*^+l) 
ax  nx 

=  3(a;-+l)--  2x,  by  II.  and  VII., 

=  6x{x^+iy. 

11 .  Find  tlie  derivative  with  respect  to  it'  of  (.c  + 1 )  (ar  —  2) . 

=  (a;  +  l)2.^  +  (a;2-2) 
=2af+2x  +  x--2 
=  3a-2+2x-2. 

Find  the  derivatives  with  respect  to  x  of  the  following  : 

12.  (3a^-l)^  17.    {l-x){2-3x'){l  +  x'). 

13.  {ax'+bx  +  cy.  18.    {x  +  'i)\x;'-2x  +  3). 

14.  x\x'-3).  '19.    (x'-2Y(x-^iy. 

15.  (2x-3)(3x  +  4).  20.    (4a;  + 5)^(4  -  5x)'- 

16.  {x-\-2){x  +  3)(x-5).    21.    (aj-+x-l)-(ar-x+l)l 

663.   Successive  Differentiation. 

If  u  is  any  function  of  x,  the  derivative  of  the  derivative 
of  u  is  called  the  Second  Derivative  of  u  ivith  resjject  to  x, 

and  is  represented  by  -—  u. 
dx^ 

The  derivative  of  the  second  derivative  of  u  is  called  the 

Third  Derivative  of  u  loith  respect  to  x,  and  is  represented  by 

. —  w;  etc. 
dx"    ' 

1.    Find  the  successive  derivatives  with  respect  to  x  of 

3x«-9a;--12x  +  2. 

We  have,     —  (3  a;^  -  9 a;^  -  12  a;  +  2)  =  9  x'  -  18  x  -  12. 
dx 


478  COLLEGE  ALGEBRA.   . 

— ,  (3x^  -  Qx"  -  12x  +  2)  =  18x  -  18. 
—ASx"-  dx'  -  12x  +  2)  =  18. 

^(3af'-9a;2-12a;  +  2)  =  0;  etc. 

Note.  It  will  be  understood  hereafter  that  when  we  speak  of  the 
derivative  of  a  function  of  x,  the  first  derivative  is  meant. 

EXAMPLES, 
rind  the  successive  derivatives  with  respect  to  03  of : 

2.  2a:2_,_^_,_i^  5^   x'-x^- 3x^+7. 

3.  of-Sx'  +  ix.  6.    2a;^  +  9a;^-21x. 

4.  3a;^  +  8a.'5-12ar^.  7.    5x^  -  Ax' ^Sx" -2. 

MULTIPLE  ROOTS. 

664.  If  an  equation  has  two  or  more  roots  equal  to  a 
(Art.  633,  Note),  a  is  said  to  be  a  Multiple  Root  of  the 
equation. 

In  the  above  case,  a  is  called  a  double  root,  tri2)le  root, 
quadnqyle  root,  etc.,  according  as  the  equation  has  two  roots, 
three  roots,  four  roots,  etc.,  equal  to  a. 

665.  Let  the  equation 

i^o X"  +  Pi x"~'^  -\ \-li„-iX+  p„  =  0  (1 ) 

have  m  roots  equal  to  a. 

By  Art.  636,  the  first  member  can  be  put  in  the  form 

(^x-arf{x)  (Art.  628);  (2) 

where  f{x)  is  the  product  of  the  factors  corresponding  to 
the  remaining  roots  of  (1),  and  is  therefore  an  integral 
expression  of  the  {n  —  m)th  degree  with  respect  to  x. 


THEORY   OF   EQUATIONS.  479 

By  Art.  660,  V.,  the  derivative  of  (2)  with  respect  to  x  is 

or,      {x  -  a)%{x)  -\-m{x-  ay-'f{x),  by  Art.  660,  VI.  (3) 

Note.  The  derivative  of  f(x)  witli  respect  to  x  is  usually  denoted 
by  /i(x). 

It  is  evident  that  the  expression  (3)  is  divisible  by 
(x  —  a)"*"^ ;  and  therefore  the  equation  formed  by  equating 
it  to  zero  will  have  m  —  1  roots  equal  to  a. 

Hence,  if  any  equation  of  the  form  (1)  has  m  roots  equal  to 
a,  the  equation  formed  by  equating  to  zero  the  derivative  of  its 
first  viemher  ivill  have  m  —  1  roots  equal  to  a. 

666.  It  follows  from  Art.  665  that,  to  determine  the 
existence  of  multiple  roots  in  an  equation  of  the  form 

.       2hx''+Pi^"'^-\ \-Pn-i^+Pn  =  0, 

we  proceed  as  follows  : 

Find  the  H.C.F.  of  the  first  member  and  its  derivative. 

If  there  is  no  H.C.F.,  there  can  be  no  multiple  roots. 

If  there  is  a  H.C.F,  by  equating  it  to  zero  and  solving  the 
resulting  equation,  the  required  roots  may  be  obtained. 

The  number  of  times  that  each  root  occurs  in  the  given 
equation  exceeds  by  one  the  number  of  times  that  it  occurs 
in  the  equation  formed  from  the  H.C.F. 

1.    Find  all  the  roots  of 

af +  »"- Gar''- 5ar  + 16x4- 12  =  0.  (1) 

The  derivative  of  the  first  member  is 

5x^  +  4.^3  -21 X--  lOx  +  16. 

The  H.C.F.  of  this  and  the  first  member  of  (1)  is  Qi?-x-2. 
Solving  the  equation  cc-—  ic  —  2  =  0,  we  have  a;  =  2  or  —1. 


480  COLLEGE   ALGEBRA. 

Hence  the  multiple  roots  of  the  given  equation  are  2,  2, 
—  1,  and  —  1. 

Dividing  12  by  minus  the  product  of  2,  2,  —  1,  and  —  1, 
the  remaining  root  is  —  3  (Art.  639). 

Therefore  the  roots  of  (1)  are  2,  2,  -  1,  -  1,  and  -  3. 

EXAMPLES. 
Find  all  the  roots  of  each  of  the  following : 

2.  af5  + 3  0^2-24  3^  +  28  =  0. 

3.  x''-4:x'-llx-C^  =  0. 

4.  8c(^  +  4x'-66x  +  63  =  0. 

5.  a;4  +  6a^  +  a^- 24a; +  16  =  0. 

6.  x*  +  7x^  +  9a^-27x-54:  =  0. 

7.  x'-7x'-i-2a^-\-12x-8  =  0. 

8.  a^''-6r^-28»2  +  120.'c  +  288  =  0.    ' 

667.  An  equation  of  the  form  a;"  —  a  =  0  can  have  no 
multiple  roots ;  for  the  derivative  of  x"  —  a  is  wa;""^,  and 
a;"  —  a  and  nx"'^  have  no  common  factor  except  unity. 

Therefore  the  n  roots  of  a;"  =  a  are  all  different. 

It  follows  from  the  above  that  every  expression  has  two 
different  square  roots,  three  different  cube  roots,  and  in 
general  n  different  wth  roots. 

LOCATION  OF  THE  ROOTS. 

668.  To  find  a  siqjerior  limit  to  the  positive  roots  of  the 
eqaalion 

f{x)  =  a;"  +  27]  .-c"-^  -\ f-p„-i  x  +  p„  =  0.  (1) 

Let  p^  be  the  absolute  value  of  the  negative  coefficient  of 
greatest  absolute  value,  and  x"~'  the  highest  power  of  x 
which  has  a  negative  coefficient. 

Then  none  of  the  first  s  terms  have  negative  coefficients. 


THEORY   OF   EQUATIONS.  481 

Now  f{x)  will  be  positive  when  x  is  positive,  provided 

X"  — j>,.a;""^  —  j:»r^"^''~^  —  •••  —PrX  —p^  (2) 

is  positive ;  for  f{x)  is  equal  to  (2)  plus 

2JiX^~^  H \- 2)^-1  x"-'+^  +  (p,  -\-p,)  X""  -\ h  (Pn  +Pr), 

a  positive  quantity. 

We  may  write  (2)  in  the  form 

.-c"  —  Pr  (a;""'  +  x"-'-^  H +  cc  +  1), 

or,  X-  _^x--^+^  -  1  ^^^^  -^^^^  ^3^ 

cc  —  1 

Then  /(cc)  will  be  positive  when  x  is  positive,  provided 
(3)  is  positive. 
But  if  x  is  greater  than  unity,  (3)  is  greater  than 

a»-p, -•  (4) 

x  —  1 

Therefore  for  values  of  x  greater  than  unity,  f{x)  will  be 
positive  if  (4)  is  positive;  or,  if  (cc  —  l)£c"— 2>,.a;"^'+'  is 
positive ;  or,  if  {x  —  1)  x'-'^  —p,  is  positive. 

But  if  X  is  greater  than  unity,  {x  —  1)  x'~'^  —p^  is  greater 
than  {x  —  1)  (cc  —  1)'"^  —  pr,  or  (a;  —  1) '  —  p^ ;  hence  for  values 
of  X  greater  than  unity,  f{x)  will  be  positive  if  {x  —  iy—p^ 
is  positive  or  zero;  or,  if  (aj  — 1)*  is  greater  than  or  equal 
to  Pr ;  or,  if  a?  —  1  is  greater  than  or  equal  to  '^Pr- 

That  is,  for  all  values  of  x  greater  than  or  equal  to 
1  +  -\fpri  /(^')  is  positive. 

Hence,  no  root  of  (1)  can  equal  or  exceed  1  +  -{/Pr\  that 
is,  1  +  -^r  is  a  superior  limit  to  the  positive  roots. 

Example.   Eind  a  superior  limit  to  the  positive  roots  of 
CF*- 19x2-460;+ 120  =  0. 

Here,  p^  =  46,  and  w  —  s  =  2 ;  whence,  s  =  2. 

Then  a  superior  limit  to  the  positive  roots  is  1  +  V46,  or 
7.78  approximately. 


482  COLLEGE   ALGEBRA. 

Note.  In  applying  the  principles  of  Art.  668,  the  term  independent 
of  X,  in  the  equation  must  be  considered  as  the  coefficient  of  x^. 

669.  By  changing  the  signs  of  the  alternate  terms  of  an 
equation  beginning  with  the  second,  an  equation  is  formed 
Avhose  roots  are  those  of  the  first  with  contrary  signs 
(Art.  643). 

The  superior  limit  to  the  positive  roots  of  the  trans- 
formed equation,  obtained  as  in  Art.  668,  with  its  sign 
changed,  will  evidently  be  an  inferior  limit  to  the  negative 
roots  of  the  given  equation. 

Example.    Find  an  inferior  limit  to  the  negative  roots  of 

cc^  +  S.T^  +  lire -13  =  0.  (1) 

Changing  the  signs  of  the  alternate  terms  beginning  with 
the  second,  we  have 

a;* +  5^2-11.^-13  =  0.  (2) 

Here,  p^  =  13,  and  ?i  —  s  =  1 ;  whence,  s  =  3. 
Then  a  superior  limit   to  the  positive   roots  of   (2)   is 

1  +  Vl3,  and  therefore  an  inferior  limit  to  the  negative 

roots  of  (1)  is  -  (1  +  V\3>^. 

EXAMPLES. 

670.  Find  a  superior  limit  to  tlio  positive  roots,  and  an 
inferior  limit  to  the  negative  roots,  of : 

1.  ic3_2ar'-3a;  +  7  =  0.         4.    x' -1  x^ -?,^ +  ^  =  Q). 

2.  af'- 5x2^.2.^4.9  =  0.         5.    u;^  +  6;c-- 3a;- 11  =  0. 

3.  a;^  +  3.'B3-4a;-10  =  0.       6.    a;'*  +  2 .x-^ - 5 a.-^  +  7  =  0. 

671.  Jf  two  real  numbers,  a  and  b,  not  roots  of  the  equation 

f(x)  =  a;"  +  2h  ^""^  H H  Pn^i  x+Pn  =  0, 

when  substitided  for  x  in  f(x),  give  results  of  opposite  sign, 
an  odd  number  of  roots  off{x)  —  0  lie  between  a  arid  b. 


THEOKY  OF   EQUATIONS.  483 

Let  a  be  algebraically  greater  than  b. 

Let  d,  ...,  g  be  the  real  roots  of  f{x)  =  0  lying  between 
a  and  b,  and  h,  ...,lc  the  remaimng  real  roots. 
Then  by  Art.  G36, 
f(x)  =  {x-d)...{x-g).{x-h)...ix-k).F{x),       (1) 

where  F{x)  is  the  product  of  the  factors  corresponding  to 
the  imaginary  roots  of /(.«;)  =  0. 

Substituting  a  and  b  for  x  in  (1),  we  have  (Art.  628), 
f{a)  =  {a-d)...{a-g)-{a-h):.{a-k).F{a), 
and      f{b)  ^(b-d)--{b-g)-{b-h)---{b-k)-F{b). 

Since  each  of  the  quantities  d,  ...,  g  is  less  than  ((  and 
greater  than  b,  each  of  the  factors  a  —  d,  ...,  a  — gin  posi- 
tive, and  each  of  the  factors  b  —  d,  ...,  b  —  g  i's,  negative. 

Again,  since  none  of  the  quantities  A,  ...,  k  lie  between  a 
and  b,  the  expression  {a—  h)---(a  —  k)  lias  tlie  same  sign 
as  {b-h)--{b-k). 

Also,  F{a)  and  F{b)  are  positive ;  for  the  product  of  the 
factors  corresponding  to  a  pair  of  conjugate  imaginary  roots 
is  positive  for  every  real  value  of  x  (Art.  641,  Note). 

But  by  hypothesis, /(a)  and/(&)  are  of  opposite  sign. 

Hence  the  number  of  factors  b  —  d,  ...,b  —  g  must  be  odd; 
that  is,  an  odd  number  of  roots  lie  between  a  and  b. 

If  the  numbers  substituted  differ  by  unity,  it  is  evident 
that  the  integral  part  of  at  least  one  root  is  known. 

672.   1.  Locate  the  roots  of  x'^  +  ^'  —  6x  —  7  —  0. 

By  Descartes'  Rule  (Art.  653),  the  equation  cannot  have 
more  than  one  positive,  nor  more  than  two  negative  roots. 

The  values  of  the  first  member  for  the  values  0,  1,  2,  3, 
—  1,  —  2,  and  —  3  of  .x  are  as  follows  : 

a;  =  0;   -7.      .x'  =  2;   -7.      a;  =  -l;   -L      a;  =  -3;   -7 
x=1;   -11.     x  =  3;  IL         x  =  -2;   1. 


484  COLLEGE    ALGEBRA. 

Since  the  sign  of  the  first  member  is  —  when  x  =  2,  and 
-f-  when  X  =  3,  one  root  lies  between  2  and  3. 

The  others  lie  between  —  1  and  —  2,  and  —  2  and  —  3, 
respectively. 

The  integral  parts  of  the  roots  are  2,  —  1,  and  —  2. 

Note.  In  locating  roots  by  the  above  method,  first  make  trial  of 
the  numbers  0,  1,  2,  etc.,  continumg  the  process  until  the  number 
of  positive  roots  determined  is  the  same  as  has  been  previously  indi- 
cated by  Descartes'  Rule. 

Thus,  in  Ex.'  1,  the  equation  cannot  have  moi'e  than  one  positive 
root ;  and  when  one  has  been  found  to  lie  between  2  and  3,  there  is  no 
need  of  trying  4,  or  any  greater  positive  number. 

The  work  may  sometimes  be  abridged  by  finding  a  superior  limit  to 
the  positive  roots,  and  an  inferior  limit  to  the  negative  roots  of  the 
given  equation  (Arts.  068,  669) ;  for  no  number  need  be  tried  which 
does  not  fall  between  these  limits. 


EXAMPLES. 
Locate  the  roots  of  the  following  equations : 
',    2.    x^- 5x^+3  =  0.      »     4.    .'B3^8.'c2-9a;-12  =  0. 
3.    x'*  — 8a;- +  15  =  0.  5.    x*  —  5x^  +  x' -j-13x —  7  =  0^ 

6.  Prove  that  the  equation  x'^  —  x-  +  2x  —  l  =  0  has  at 
least  one  root  between  0  and  1. 

7.  Prove  that  the  equation  x*  —  2x^  —  3  a;-  -+-  a;  —  2  =  0  has 
a  root  between  —  1  and  —  2,  and  at  least  one  between  2  and  3. 

8.  Prove  that  the  equation  x*  +  x^-i-2x-  —  x  —  1  =  0  has  a 
root  between  0  and  1,  and  at  least  one  between  0  and  —  1. 

673.   Location  of  Roots  by  Synthetic  Division  (Art.  650). 
Let  Q  denote  the  quotient,  and  R  the  remainder  obtained 
by  dividing  the  expression 

a;"  +27i^'"  '  H \-P„-iX  +Pn  (1) 

hy  X  —  a ;  then, 

X"  +2h X"  '  +  •••+  ;>„  ,  X  +  p„  =  Q  (x  -  a )  (- 11.       (2) 


THEORY   OF   EQUATIONS.  485 

Putting  X  =  a  in  (2),  we  obtaiu 

a"  +  2h  rt"  "'  +  •••  +Pn~i «  +  pH  =  -K- 

That  is,  the  value  of  (1)  with  a  written  in  place  of  x  is 
equal  to  the  remainder  obtained  by  dividing  (1)  by  a;  —  a. 

It  follows  from  the  above,  in  connection  with  Art.  671, 
that  if  a  and  b  are  real  numbers,  not  roots  of  the  equation 

/  (.^•)  =  X"  +  /)i  x"-^-{ f-  p„  _  1 .1-  +  p,  =  0, 

and,  when  f{x)  is  divided  by  x  —  a  and  x  —  b,  the  re- 
mainders are  of  opposite  sign,  an  odd  number  of  roots  of 
/(.t)  =  0  lie  between  a  and  b. 

The  remainders  may  be  obtained  by  Synthetic  Division. 

1.    Locate  the  roots  oioif-{-x-  —  ox  —  A  =  0. 

By  Descartes'  Rule,  the  equation  cannot  have  more  than 
one  •i^)ositive,  nor  more  than  two  negative  roots. 

Dividing  x'^-j-ay^—5x—4:  by  x,  the  remainder  is  —  4.         (o) 

Dividing  the  first  member  successively  by  x  —  1,  x  —  2, 
x  —  S,  x+1,  ic  +  2,  and  x  +  3,  we  have 

1   +1   -5  -4  1-1   (7) 

- 1       0  5 

0   -0  ^ 

1   +1   _5  _4  1-2   (8) 

—  2       2  () 
^  ^  ~2 

1   +1   -5  -4  1_--^  (9) 

—  3       6   —  3 

In  (5)  and  (6),  the  remainders  are  —2  and  +17,  re- 
spectively ;  hence  one  root  lies  between  2  and  3. 

In  (3)  and  (7),  the  remainders  are  —  4  and  + 1,  respec- 
tively ;  hence  a  root  lies  between  0  and  —  1. 

In  like  manner,  a  root  lies  between  —  2  and  —  3. 


1 

+  1 

—  5 

-4  LI   (4) 

1 

o 

-3 

2 

^ 

—  7 

1 

+  1 

—  5 

-^[2  (5) 

2 

6 

'> 

3 

1 

« 

1 

+  1 

^5 

-4L3  (G) 

3 

12 

21 

4 

7 

17 

486  COLLEGE   ALGEBRA. 

Note  1.  The  above  process  is  nothing  more  than  a  convenient 
way  of  applying  the  test  of  Art.  671.  It,  has  moreover  the  advantage 
over  the  method  of  direct  substitution  that,  when  the  integral  part  of 
a  root  has  been  found,  the  work  performed  is  identical  with  the  first 
part  of  Horner's  method  (Art.  718)  for  determining  additional  root- 
figures  ;  thus,  in  Ex.  1,  the  work  m  (5)  is  identical  with  the  first  three 
lines  of  the  determination  by  Horner's  method  of  the  root  of  the  given 
equation  lying  between  2  and  3. 

Note  2.  The  Note  to  Art.  672  applies  with  equal  force  to  tne 
method  of  Art.  673. 

EXAMPLES. 

Locate  the  roots  of  the  following  equations : 

2.  ar^-5ar  +  2a;  +  6  =  0.        4.    x' -15ar -{-3x +  U  =  0. 

3.  ar'  +  2ar'-x-l  =  0.  5.    x*  +  6  ar'' -  42  .x  -  44  =  0. 

6.  Prove  that  the  equation  a;^  +  3.x  —  5  =  0  has  one  root 
between  1  and  2. 

7.  Prove  that  the  equation  x* —  3x^ -{- 6x- -{-x —  1  =  0 
has  one  root  between  0  and  —  1,  and  at  least  one  root 
between  0  and  1. 

674.  The  methods  of  Arts.  672  and  673,  though  simple 
in  principle  and  easy  of  apj^lication,  are  not  sulticient  to 
deal  with  every  problem  in  location  of  roots. 

Let  it  be  ret^uired,  for  example,  to  locate  the  roots  of 
x!^  +  3x^-\-2x  +  l  =  0. 

We  know  by  Art.  642  that  the  *^uation  has  at  least  one 
real  root. 

By  Descartes'  Eule,  the  equation  has  no  positive  root. 

By  Art.  669,  —  4  is  an  inferior  limit  to  the  negative  roots. 

Putting  X  equal  to  0,  —  1,  —  2,  and  —  3,  the  correspond- 
ing values  of  the  first  member  are  1,  1,  1,  and  —  5. 

Therefore  the  equation  has  either  one  root  or  three  roots 
between  —2  and  —3;  but  the  methods  already  given  are 
iusuincicut  to  determine  which. 


THEORY   OF  EQUATIOXS.  487 

Sturm's  Theorem  (Art.  675)  affords  a  method  for  deter- 
mining completely  the  number  and  situation  of  the  real 
roots  of  an  equation.  It  is  more  difficult  of  application 
than  the  methods  of  Arts.  672  and  673,  and  should  be  used 
only  in  cases  which  the  latter  cannot  resolve. 

675.   Sturm's  Theorem. 

Let        /(.^•)  =  x"  -j-2),x''~'  +  •••  +Pn-i3^+Pn  =  0  (1) 

be  an  equation  from  which  the  multiple  roots  have  been 
removed  (Art.  666). 

Let/i(x)  denote  the  derivative  oi  f(x)  with  respect  to  x 
(Art.  658). 

Dividing /(cc)  by /i(a;),  we  shall  obtain  a  quotient  Qi,  with 
a  remainder  of  a  degree  lower  than  that  of /(.i-). 

Denote  this  remainder,  with  the  sign  of  each  of  its  terms 
changed,  ^J  fXx),  and  divide  f{x)  hy  f{x),  and  so  on;  the 
operation  being  precisely  the  same  as  that  of  finding  the 
H.C.F.  of /(a;)  and/i(aj),  except  that  the  signs  of  the  terms 
of  each  remainder  are  to  be  changed,  while  no  other  changes 
of  sign  are  permissible. 

Since /(a)  =  0  has  no  multiple  roots, /(cc)  and/i(a;)  have 
no  common  divisor  except  unity  (Art.  Q>QQ)) ;  and  we  shall 
finally  obtain  a  remainder /^  (a?)  independent  of  x. 

The  expressions  f{x),  fi{x),  fix),  ...,  fn(x)  are  called 
Sturm's  Functions. 

The  successive  operations  are  represented  as  follows  : 

f{^)=QiM^)-f2{x),  (2) 

f{x)=QMx)-f{x),  (3) 

f2(x)=QJsix)-f{x),  (4) 


fnMx)=Qjj„^,(x)-fXx). 
We  may  now  enunciate  Sturm's  Theorem. 


488  COLLEGE   ALGEBRA. 

If  two  real  numbers,  a  and  b,  are  substituted  in  place  of  x 
in  Sturm/s  Functions,  and  the  signs  noted,  the  difference  be- 
tween the  number  of  variations  of  sign  (Art.  653,  Note  1)  in 
the  first  case  and  that  in  the  second  is  equal  to  the  number  of 
real  roots  off{x)  =  0  lying  between  a  and  b. 

The  demonstration  of  the  theorem  depends  upon  the  fol^ 
lowmg  prmciples : 

I.  Tivo  consecutive  functions  cannot  both  become  0  for  the 
same  value  of  x. 

For  if,  for  any  value  of  x,  f{x)  =  0  and  fo{x)  =  0,  then  by 
(3),  fsix)  =  0;  and  since  f{x)  =  0  and  f{x)  =  0,  by  (4), 
f[x)  =  0;  continuing  in  this  way,  we  shall  finally  have 
f,,{x)  =  0. 

But  by  hypothesis, /„ (a;)  is  independent  of  x,  and  conse- 
quently cannot  become  0  for  any  value  of  x. 

Hence  no  two  consecutive  functions  can  become  0  for  the 
same  value  of  x. 

II.  If  any  function,  except  f{x)  and  fn(x),  becomes.  0  for 
anji  value  of  x,  the  adjacent  functions  have  opposite  signs  for 
this  value  of  x. 

For  if,  for  any  value  of  a;,  f{o:)  =  0,  then,  by  (3),  we  must 
have/i(a;)  =  —fi{x)  for  this  value  of  x. 

Therefore  fx{x)  and  f{x)  have  opposite  signs  for  this 
value  of  X ;  for,  by  I.,  neither  of  them  can  equal  zero. 

III.  Let  c  be  a  root  of  the  equation  fr{x)  =  0,  where  /,(.r ) 
is  any  function  except /(x)  and/„(a;). 

By  11.,  fr-i{x)  and/.+j(£c)  have  opposite  signs  when  x=c. 

Now  let  h  be  a  positive  quantity,  so  taken  that  no  root 
of  fr^i{x)  =  0  or /,.^i(.r)  =  0  lies  between  c  —  h  and  c  +  h. 

Then  as  x  changes  from  c  —  h  to  c  +  h,  no  change  of  sign 
takes  place  in  fr.i{x)  or  f+i{x)  ;  while  f{x)  reduces  to 
zero,  and  changes  or  retains  its  sign  according  as  the  root  c 
occurs  an  odd  or  even  number  of  times  in/,.(a;)=0. 


TFIEORY   OF   EQUATIONS.  489 

Therefore,  for  values  of  x  between  c  —  h  and  c,  and  also 
for  values  of  x  between  c  and  c  +  h,  the  three  functions. 
fr-i{x),  f,.{x),  and  fr+i{x)  present  one  permanence  and  one 
variation. 

Hence,  as  x  increases  from  c  —  k  to  c  -\-  h,  no  change  occurs 
in  the  number  of  variations  in  the  functions  fr-i{^-),  f,i^'), 
and./^^.i(a;) ;  that  is,  no  change  occurs  in  the  number  of 
variations  as  x  increases  through  a  root  of /.(x)  =  0. 

IV.  Let  c  be  a  root  of  the'equation /(.!■)  =  0  ;  and  let /« 
be  a  positive  quantity,  so  taken  that  no  root  oij\{x)  =  0 
lies  between  c  —  h  and  c  +  h. 

Then  as  x  increases  from  c  —  /i  to  c  -f  h,  no  change  of  sign 
takes  place  in  fi{x) ;  while  f{x)  reduces  to  zero,  and  changes 
sign. 

Putting  x  =  c  —  h  in  (1),  we  obtain 

/(c  -  k)  =  (c  -  hY+p,{c  -  hy  '  +  •••  +  p„_i(c  -  h)+p^. 

Expanding  by  the  Binomial  Theorem,  and  collecting  the 
terms  involving  like  powers  of  h,  we  have 

/(c  -  h)  =  c"  +;?i  c"-^  +  •  •  •  +  p„  -1  c  -h  21. 

—  h[nc"  '^  +  (n  —  1  )29, c"~-  +  ■■■  +  p„_i] 
+  terms  involving  /i^,  /i'',  ...,  h". 

But  since  c  is  a  root  of /(«)  =  0,  we  have  by  (1), 

c^+iJ^c"  ^-] \-Pn-iC+P„  =  0. 

Also,  it  is  evident  that  the  coefficient  of  —  /*  is  the  value 
of /i(x)  when  c  is  substituted  in  place  of  x;  therefore, 

/(c  —  /i)  =  —  /i/i(c)  +  terms  involving  h-,  h^,  ...,  h'\    (5) 

In  like  manner  it  may  be  shown  that 

/(c  +  7i)  =  -{-  /i/j|(c)  +  terms  involving  /r,  h^,  ...,  h".    (6) 

Now  if  h  is  taken  sufficiently  small,  the  signs  of  the  sec- 
ond members  of  (5)  and  (6)  will  be  the  same  as  the  signs 
of  their  first  terms,  —hfi(c)  and  +/'/i(c),  respectively. 


490  COLLEGE   ALGEBRA. 

Hence,  if  h  is  taken  sufficiently  small,  the  sign  of /(c  —  h) 
will  be  contrary  to  the  sign  of /,(c),  and  the  sign  of /(c  +  h) 
will  be  the  same  as  the  sign  of /i(c). 

Therefore,  for  values  of  x  between  c  —  h  and  c,  the  func- 
tions f{x)  and  fi{x)  present  a  variation,  and  for  values  of 
X  between  c  and  c-\-h  they  present  a  permanence. 

Hence  a  variation  is  lost  as  x  increases  through  a  root 
of/(a;)  =  0. 

We  may  now  demonstrate  Sturm's  Theorem ;  for  as  x 
increases  from  h  to  a,  supposing  a  algebraically  greater  than 
b,  a  variation  is  lost  each  time  that  x  passes  through  a  root 
of  f{x)  =  0,  and  only  then ;  for  when  x  passes  through  a 
root  of  f^(x)  =  0,  where  fr{x)  is  any  function  except  f(x) 
and/„  (x),  no  change  occurs  in  the  number  of  variations. 

Hence  the  number  of  variations  lost  as  x  increases  from 
&  to  a  is  equal  to  the  number  of  real  roots  of  f{x)  =  0  in- 
cluded between  a  and  b. 

676.  It  is  customary,  in  applications  of  Sturm's  Theorem, 
to  speak  of  the  substitution  of  an  indefinitely  great  number 
for  X,  in  an  expression,  as  substituting  /y:!  for  x. 

The  substitution  of  +  co  and  —  oo  for  x  in  Sturm's  Func- 
tions determines  the  number  of  real  roots  of  f(x)  =  0. 

The  substitution  of  -f  oo  and  0  for  x  determines  the  num- 
ber of  positive  real  roots,  and  the  substitution  of  —  co  .and  0 
determines  the  number  of  negative  real  roots. 

677.  Since  Sturm's  Theorem  determines  the  number  of 
real  roots  of  an  equation,  the  number  of  imaginary  roots 
also  becomes  known  (Art.  633). 

678.  If  a  sufficiently  great  number  is  substituted  in 
place  of  X  in  the  expression 

F{x)  =  2hx"-\-PiX^-'^-\----  +Pn^iX+p„, 
the  sign  of  the  result  will  be  the  same  as  the  sign  of  its 
first  term,  poX". 


THEORY   OF   EQUATIONS.  491 

It  follows  from  tlie  above  that : 

7/"  +  c»  is  substituted  in  place  of  x  in  F{x),  the  sign  of  the 
result  is  the  same  as  the  sign  of  its  first  term. 

Jf  —yD  is  substituted  in  x)lace  of  x  in  F{x),  the  sign  of  the 
result  is  the  same  as,  or  contrary  to,  the  sign  of  its  first  term, 
according  as  the  degree  of  F{x)  is  even  or  odd. 

679.  In  the  process  of  finding  /^(a;),  fi{x),  etc.,  any 
positive  numerical  factor  may  be  omitted  or  introduced  at 
pleasure ;  for  the  sign  of  the  result  is  not  affected  thereby. 

In  this  way  fractions  may  be  avoided. 

680.  1.  Determine  the  number  and  situation  of  the  real 
roots  of 

/(x)  =  a;3  -  63.-2  +  5a;  +  13  =  0. 

Here,  f  {x)  =  3  a;^  -  12  a;  +  5. 

Multiplying  f{x)  by  3  in  order  to  make  its  first  term 
divisible  by  33;^,  we  have 

3a;2  -  12a;  +  5)3ar^  -  18a;2  +  15.a;  +  ,39(a;  -  2 

3a;"  — 12a;2+    5a; 


-  6a;2  +  10a;  +  39 

-  6a;- +  24a; -10 

7) -14a; +  49 
-    2a;+    7 

3a;2-12a;+    5 

2 

. /,(a;)  =  2a;-7. 

2a;-7)6ar^-24a;  +  10(3a; 
Qx'-llx 

-    3a; +  10 

2 

-  6a;  +  20(-3 

-  6a; +  21 

-    1  .-. /3(a;)  =  l. 


492  collegp:  algebra. 

Substituting  -co  for  x  in  f{x),  fi(x),  /^{x),  and  fs{x), 
the  signs  are  — ,  -\-,  — ,  and  -\-,  respectively  (Art.  678)  ;  sub- 
stituting 0  for  X,  the  signs  are  +,  +,  — ,  +,  respectively; 
and  substituting  +  a>  for  x,  the  signs  are  all  + . 

Hence  the  roots  of  the  equation  are  all  real,  and  two  of 
them  are  positive  and  the  other  negative. 

We  now  substitute  various  numbers  to  determine  the 
situation  of  the  roots  : 


A^) 

M^) 

Mx) 

fz{^) 

a;  =  —  CO, 

— 

+ 

— 

+ 

3  variations. 

x  =  -2, 

— 

+ 

— 

+ 

3  variations. 

x  =  -l, 

+ 

+ 

— 

+ 

2  variations. 

X=Q, 

+ 

+ 

— 

-f 

2  variations. 

x  =  l, 

+ 

— 

— 

+ 

2  variations. 

x  =  2, 

+ 

— 

— 

+ 

2  variations. 

X  =  3, 

+ 

- 

- 

+ 

2  variations. 

a;=4, 

+ 

+ 

+ 

+ 

no  variation. 

a  =  co, 

+ 

+ 

+ 

+ 

no  variation. 

We  then  know  that  the  equation  has  one  root  between 
—  1  and  —  2,  and  two  roots  between  3  and  4. 

Note  1.  In  substituting  the  numbers,  it  is  best  to  work  from  0  in 
either  direction,  stopping  when  the  number  of  variations  is  the  same  as 
has  been  previously  found  for  -f-co  or  —go,  as  the  case  may  be. 

2.  Determine  the  number  and  situation  of  the  real  roots 
of 

/(.T)  =  4«''-2a;-5  =  0. 

Here,  fiix)  =  12.'c^  —  2 ;  or,  Ga;-  —  1,  omitting  the  factor  2. 

^x^-2x-    5 
3 


6c^-l)12x--()x-l5{2x 
12a;'' -2a; 


-4«-15        .-. /,(a;)  =  4x  +  15. 


THEORY   OF   EQUATIONS.  493 

6x^-1 

2 


4a; +  15)  12 a;--    2(30; 
12a;-  +  45£C 


-45x-      2 
4 
-180x-     8(-45 
- 180  X- 675 


667        .-. /3(a;)  =  -667. 

Note  2.  The  last  step  in  the  division  may  be  omitted  ;  for  wc  only 
need  to  know  the  sig7i  of /j  (x)  ;  and  it  is  evident  by  inspection,  when 
tne  remainder  —  45  a;  —  2  is  obtained,  that  the  sign  of  /j  (x)  will  be  — . 

/(^)  M^)  f2{x)  f,{x) 

x  =  —  00,     —  +  —  —     2  variations. 

cc  =  0,          —  —  +  —     2  variations. 

x  =  l,          —  +  +  —     2  variations, 

a;  =  2,          +  +  +  —     1  variation. 

x  =  oo,         +  +  +  —     1  variation. 

Therefore  the  equation  has  a  real  root  between  1  and  2, 
and  two  imaginary  roots. 

EXAMPLES. 

Determine  the  number  and  situation  of  the  real  roots  of ; 

3.  ar^_4a;2-4a;+12  =  0.         7.    a;*- 12a;2  + 12cc- 3  =  0. 

4.  a;^  +  5a;  +  2  =  0.  8.    2a;*- 3a;-  +  3.r  -  1  =  0. 

5.  a;3  +  3a;2-9a;-4  =  0.  9.    a;*+2ar^-6a;2-8a;+9=0. 

6.  ar''-4a;2- 10a; +  41  =  0.     10.    a;''  + 4a;3 +  3a;  +  27  =  0. 

681.  As  X  increases  from  —  oo  to  +go,  f{x)  and  /i(a;) 
change  signs  alternately,  for  they  are  ahvays  unlike  in  sign 
just  before /(a;)  changes  sign  (Art.  675,  IV.);  hence,  if  the 
roots  of /(x-)  =  0  and/i(a;)  =  0  are  all  real,  a  root  of /i(a;)  =0 
lies  between  every  two  adjacent  roots  of /(a;)  =  0. 


P,(-a,h) 

Y 

P,(a,h) 

h\ 

i 

„'      1         o 

a      j      . 

^    \n 

0 

m\    ' 

h\ 

Z, 

P,{'a,-b) 

Y' 

1 

Pi  {.a,-h 

494  COLLEGE   A1>GEBRA. 

GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 

682,  Rectangular  Co-ordinates. 

Let  Fi  be  any  point  in  the  plane  of  tlie  lines  XX'  and 
YY',  intersecting  at  right  angles 
at  0. 

Draw    Pi  31  perpendicular    to 
XX'. 

Then  03f  and  A  M  are  called 
the    rectangular   co-ordinates    of 
-b)    PjI    OM  is   called  the  abscissa, 
and  Pi  M  the  ordinate. 
The  lines  of  reference,  XX'  and  YY',  are  called  the  axes 
of  X  and  Y,  respectively,  and  0  is  called  the  origin. 

It  is  customary  to  express  the  fact  that  the  abscissa  of 
a  point  is  a,  and  its  ordinate  b,  by  saying  that,  for  the 
point  in  question,  x  =  a  and  y  =  b;  or,  more  concisely  still, 
we  may  refer  to  the  point  as  "the  point  (a,  6),"  where  the 
first  term  in  the  parenthesis  is  understood  to  be  the  abscissa, 
and  the  second  term  the  ordinate. 

683.  If,  in  the  figure  of  Art.  682,  OM=ON=a,  and 
P,  7^4  and  P2P3  are  drawn  perpendicular  to  XX'  so  that 
PiM=P.,N=P^N=PiM=b,  the  points  Pi,  Po,  P3,  and  P, 

will  have  the  same  co-ordinates,  {a,  b). 

To  avoid  this  ambiguity,  abscissas  measured  to  the  right 
of  0  are  considered  positive,  and  to  the  left,  negative;  and 
ordinates  measured  above  XX'  are  considered  positive,  and 
beloio,  negative. 

Then  the  co-ordinates  of  the  points  will  be  as  follows : 
Pi,  (a,  h)  ;  P,  (-a,  h) ;  P,^{-a,-h);  P„  (a,  -  b). 

If  a  point  lies  upon  XX',  its  ordinate  is  zero ;  and  if  it 
lies  upon  YY',  its  abscissa  is  zero. 

The  co-ordinates  of  the  origin  are  (0,  0). 


THEORY   OF   EQUATIONS. 


495 


To  plot  a  point  when  its  co-ordinates  are  given,  lay  off 
the  abscissa  to  the  right  or  left  of  0,  according  as  it  is 
positive  or  negative,  and  then  draw  a  perpendicular,  equal 
in  length  to  the  ordinate,  above  or  below  XX'  according  as 
the  ordinate  is  positive  or  negative. 

Thus,  to  plot  the  point  (—3,  2),  lay  off  3  units  to  the 
left  of  0  upon  XX',  and  then  erect  a  perpendicular  2  units 
in  length  above  XX'. 


684.   Graph  of  a  Function  of  x  (Art.  213). 

Consider  the  function  x-  —  2x  —  3.  and  put  y  =  x-  —  2x  —  o. 

If  we  give  any  numerical  value  to  x,  we  may,  by  aid  of 
the  relation  y  =  x'  —  2x  —  3,  calculate  a  corresponding  value 
for  y. 

•  The  following  are  the  values  of  y  corresponding  to  the 
values  0,  1,  2,  3,  4,  -  1,  -  2,  and  -  3  of  a: : 


^; 

x  =  0, 

y  =  -o. 

E] 

x=      4, 

y=  5. 

B; 

x  =  l, 

2/  =  -4. 

F; 

x  =  -l, 

y=  0. 

C; 

x  =  2, 

y  =  -3. 

G; 

x=-2, 

y^  5. 

D; 

X  =  3, 

y=      0. 

H; 

X  =  —  3, 

2/ =  12. 

Now  let  the  above  be  regarded  as  the  co-ordinates  of  points, 
and  let  these  points  be  plotted,  as  explained  in  Art.  683. 

The  points  will  be  found  to  lie 
on  a  certain  curve,  GBE,  which  is 
called  the  Graph  of  the  given  func- 
tion. 

The  point  H  falls  without  the 
limits  of  the  figure. 

By  taking  other  values  for  x,  the 
curve  may  be  traced  beyond  E  and 
G ;  and  since  y  increases  indefi- 
nitely with  X,  the  gra])h  extends 
in  either  direction  to  an  indefinitely  great  distance  from  0. 


496 


COLLEGE    ALGEBRA. 


By  taking  values  of  x  between  those  assumed  above,  the 
curve  may  be  located  to  any  desired  degree  of  precision 
between  E  and  G. 


685.   Let   it    be    required   to   construct    the    graph    of 

Putting  y  =  2x'^  —  ox,  we  have  the 
following;  results : 


x  =  (), 

y=    0 

.=1 

—4' 

x  =  l, 

y  =  -i 

-1 

-  ! 

-1- 

5 
4' 

-i,y  = 

1. 

3 

"2'  ^^' 

9 

4° 

The  graph  of  the  function  is  the  curve  AOB. 

This  graph  also  extends  indefinitely  beyond  A  and  B. 

686.  To  determine  the  points  where  the  graph  of  any 
function  of  x  intersects  XX',  we  require  to  know  what 
values  of  x  will  make  y  equal  to  zero. 

This  may  be  effected  by  placing  the  given  function  equal 
to  zero,  and  solving  the  resulting  equation. 

It  follows  from  this  that  the  graph  of  the  first  member  of 
PoX"" -\- pix"-'^ -\ l-i5„-iic+P„  =  0 

intersects  XX'  as  many  times  as  the  equation  has  real 
unequal  roots,  and  that  the  abscissas  of  the  points  of  inter- 
section are  the  values  of  the  roots. 

Thus  in  Art.  684,  the  graph  of  cc^  —  2  a;  —  3  intersects 
XX'  twice ;  at  cc  =  3  and  a;  =  —  1.  > 

Hence,  the  equation  x^  —  2x  —  S  =  0  has  two  real  roots ; 
3  and  -  1. 


THEORY   OF   EQUATIOXS.  •     407 

Again,  in  Art.  685,  the  graph  of  2  x^  —  3  a;  intersects  XX' 

'3 

three  times ;  once  between  x  =  l  and  x=-,  once  between 

3 

x  =  —  l  and  x  = ,  and  once  at  x  =  0. 

Hence,  the  equation  2cc^  — 3^  =  0  has  three  real  roots; 

one  between  1  and  -,  one  between  —  1  and  — '-,  and  one 

2  2 

equal  to  zero. 

If  the  graph  does  not  intersect  XX'  at  all,  the  equation 
formed  by  placing  the  function  equal  to  zero  has  only 
imaginary  roots. 

687.  The  principles  of  Art.  686  may  be  used  to  locate 
the  roots  of  an  equation. 

For  if,  in  the  graph  of  f{x),  the  points  Avhose  abscissas 
are  a  and  b,  respectively,  lie  on  opposite  sides  of  XX',  an 
odd  number  of  roots  of  /(x)  =  0  lie  between  a  and  b 
(Art.  671). 

The  method  is  simply  a  graphical  representation  of  the 
process  of  Art.  672,  and  is  subject  to  the  limitations  stated 
in  Art.  674. 

EXAMPLES. 

688.  Plot  the  graphs  of  the  following  functions  : 

1.  3a; -5.  3.    x^-1.  5.    a;'' -7a; +  4. 

2.  a;-  +  4a;  +  4.     4.    x'-4:x'  +  2.     6.    a.-^  -  9 a;'^  +  23 a;  -13. 
Locate  the  roots  of  the  following  equations  : 

7.  a;3  +  4x--7  =  0.  9.    a;''  +  9a;2  + 23a;  + 17  =  0. 

8.  x^-6x'  +  7x  +  3  =  0.       10.    x'  +  x'-Sx'-x  +  l^O. 


498  COLLEGE   ALGEBRA. 

XLII.   SOLUTION  OP  HIGHER  EQUATIONS. 

COMMENSURABLE  ROOTS. 

Note.  We  shall  use  the  term  commensurable  root,  in  Chap.  XLII., 
to  signify  a  rational  root  (Art.  209)  expressed  ni  Arabic  numerals. 

689.  By  Art.  040,  an  equation  of  the  nt\\  degree  in  its 
general  form  (Art.  626),  with  integral  numerical  coeffi- 
cients, cannot  have  as  a  root  a  rational  fraction  in  its 
lowest  terms. 

Therefore,  to  find  all  the  commensurable  roots  of  such  an 
equation,  we  have  only  to  find  all  its  integral  roots. 

Again,  by  Art.  638,  the  last  term  of  an  equation  of  the 
above  form  is  divisible  by  every  integral  root. 

Hence,  to  find  alj  the  commensurable  roots,  we  have  only 
to  ascertain  by  trial  which  integral  divisors  of  the  last  term  are 
roots  of  the  equation. 

I     The  trial  may  be  made  in  three  ways  : 
V^  I^  By  actual  substitution  of  the  supposed  root. 

II.  By  dividing  the  first  member  of  the  equation  by  the 
unknown  quantity  minus  the  supposed  root  (Art.  629) ;  in 
this  case,  the  operation  may  be  conveniently  performed  by 
Synthetic  Division  (Art.  650). 

III.  By  the  Method  of  Divisors  (Art.  091). 

In  the  case  of  small  numbers,  such  as  ±1,  the  first  method 
may  be  the  most  convenient.  The  second  method  has 
the  advantage  that,  when  a  root  has  been  found,  the  process 
gives  at  once'  the  depressed  equation  (Art.  635)  for  obtain- 
ing the  remaining  roots.  If  the  number  of  divisors  is  large, 
the  third  method  will  be  found  to  involve  the  least  work. 

Considerable  work  may  sometimes  be  saved  by  finding  a 
superior  liinit  to  the  positive  roots,  and  an  inferior  limit  to 
the  negative  roots  (Arts.  668,  669) ;  for  no  number  need  bo 
tricid  which  does  not  fall  between  these  limits. 


SOLUTION    OF   HIGHER   EQUATIONS.  499 

Descartes'  Eule  of  Signs  (Art.  653)  may  also  be  advan- 
tageously employed  to  shorten  the  process. 

Any  multiple  root  should  be  removed  (Art.  666)  before 
applying  either  method. 

Example.    Find  all  the  roots  of  a;*  —  15  a;-  -|-  10  a:  -|-  24  =  0. 

By  Descartes'  Rule,  the  equation  cannot  have  more  than 
two  positive,  nor  more  than  two  negative  roots. 

By  Arts.  668  and  669,  1  +  Vl5  is  a  superior  limit  to  the 
positive  roots,  and  —(1  +Vl5)  is  an  inferior  limit  to  the 
negative  roots. 

The  integral  divisors  of  24  lying  between  1  -|-  Vl5  and 

—  (1  +  Vi5)  are  ±1,  ±2,  ±3,  and  ±  4. 

By  actual  substitution,  we  find  that  1  is  not,  and  that 

—  1  is,  a  root  of  the  equation. 

Dividing  the  first  member  by  a;  —  2,  a;  —  3,  etc.  (Art. 
650),  we  have 

1-+0 -15  +10  +24  [2  1  +0  -15  +10  +24  |_3 

2         4-22-24  3         9-18-24 

2-11-12,       0  Rem.         ~~3  ^^  ^I~8,       0  Rem. 

The  work  shows  that  2  and  3  are  roots  of  the  given 
equation ;  and  since  the  equation  cannot  have  more  than 
two  positive  roots,  these  are  the  only  positive  roots. 

The  remaining  root  may  be  found  by  dividing  24  by  the 
product  of  —  1,  2,  and  3  (Art.  639),  or  by  the  same  process 
as  above. 

Dividing  the  first  member  by  a;  +  2,  a;  +  3,  etc.,  we  have 

1+0-15+10+24  1^-2       1+0-15+10+24  \^-^ 

-2  4  _22  -64  - 4       16-4-24 

-2^11~~32l^  1^4        i        6        0 

1+0-15+10+24  1-3 

-3   9      18  -  84 

Zs  ^^  ^8  ^^ 


600  COLLEGE   ALGEBRA. 

The  work  shows  that  the  remaining  root  is  —  4. 
Thus,  the  four  roots  of  the  given  equation  are  —  1,  2,  3, 
and  —  4. 

690.  By  Art.  645,  an  equation  of  the  nth  degree  in  its 
general  form,  with  fractional  coefficients,  may  be  trans- 
formed into  another  whose  coefficients  are  integral,  that  of 
the  first  term  being  unity.       , 

The  commensurable  roots  of  the  transformed  equation 
may  then  be  found  as  in  the  preceding  article. 

Examj)le.  Find  all  the  roots  of  4ar^  -  12. t^  +  27a;  -  19  =  0. 
Dividing  through  by  the  coefficient  of  x',  we  have 

x^  —  3  X-  H =  0. 

4         4 

Proceeding  as  in  Art.  645,  it  is  evident  by  inspection 
that  the  multiplier  2  will  remove  the  fractional  coefficients ; 
thus  the  transformed  equation  is 

a:3_2.3a;2  +  22.?^-23.— =  0, 
4  4 

or,  ar"^- 6x^4- 27a;- 38  =  0;  (1) 

whose  roots  are  those  of  the  given  equation  multiplied  by  2. 

By  Descartes'  Rule,  equation  (1)  has  no  negative  root. 

The  positive  integral  divisors  of  38  are  1,  2,  19,  and  38. 

Dividing  the  first  member  by  a;  —  1,  a;  —  2,  etc.,  we  have 

1   _  6  +  27  -  38  |JL_  1   -  6  +  27  -  38  |_2^ 

1   -    5       22  2-8    ^8 

^   "^   ^16  _4       19         0 

The  work  shows  that  2  is  a  root  of  (1). 

The  remaining  roots  may  now  be  found  by  depressing 
the  equation;  it  is  evident  from  the  right-hand  operation 
above  that  the  depressed  equation  is  x^  —  4  x  +  19  =  0. 


SOLUTION   OF   HIGHER   EQUx\TTONS.  501 

Solving  this  by  the  rules  for  quadratics,  we  have 


a;  =  2  ±  V4  -  19  =  2  ±  V-lo. 
Thus  the  three  roots  of  (1)  are  2  and  2  ±V- 15. 
Dividing  by  2,  the  roots  of  the  given  equation  are 

1  and  l±j  V^^IS. 

691.  Newton's  Method  of  Divisors. 

If  a  is  an  integral  root  of  the  equation 

x"  +  PiX"-^  -\ h ;?„_^ x^  +  p^  J ic  +  p„  =  0, 

where 2?!,  ...,p„  are  integers,  then 

a"  +pia"-'  H \-Pn-2a-  +i>,.-i«  +Pn  =  0. 

Transposing,  and  dividing  by  a, 

Ph 

-  =-p„-i-i>„-2« Pja"-2-a»-i;  (1) 

Pi 
from  which  it  is  seen  that  -"  must  be  an  integer. 


We  may  write  (1)  in  the  form 

Pn 


Pi  a'' 


Representing  -^  +Pn-i  by  q„  j,  and  dividing  by  a, 

Qn-i 

from  which  it  is  seen  that  -^  must  be  an  integer. 

Proceeding  in  this  way,  it  is  evident  that,  if  a  is  a  root  of 

-^+Pn~2  „ 

the  equation,  each  of  the  quantities  or  ^^^,  .... 

—  +lh 

or  - ,  must  be  an  integer,  and  —  +  1  must  equal  0. 


502  COLLEGE   ALGEBRA. 

We  then  have  the  following  rule : 

Divide  the  last  term  of  the  equation  by  one  of  its  integral 
divisors,  and  to  the  quotient  add  the  coefficient  of  x. 

Divide  the  result  by  the  same  divisor,  and,  if  the  quotient  is 
an  integer,  add  to  it  the  coefficient  ofa^. 

Proceed  in  this  manner  with  each  coefficient  in  succession ; 
then,  if  the  divisor  is  a  root  of  the  equation,  each  quotient  will 
be  integral,  and  the  last  quotient  added  to  unity  will  equal  zero. 

If  a  fractional  quotient  is  obtained  at  any  stage,  the 
corresponding  divisor  is  not  a  root  of  the  equation. 

Example.  Find  all  the  roots  of  aj*  —  or^  —  7  x^  +  a;  +  6  =  0. 

By  Descartes'  Rule,  the  equation  cannot  have  more  than 
two  positive,  nor  more  than  two  negative  roots. 

The  integral  divisors  of  6  are  ±1,  ±2,  ±3,  and  ±  6. 

By  actual  substitution,  we  find  that  1  and  —  1  are  roots. 

We  will  next  ascertain  if  2  is  a  root;  a  convenient 
arrangement  of  the  work  is  shown  below : 

.   1   _1   _7   +1   +6  \2_ 
2       3 
-5       4 

The  operation  is  carried  out  as  follo^\^s  : 
Dividing  6  by  2,  gives  3 ;  adding  1,  gives  4. 
Dividing  4  by  2,  gives  2 ;  adding  —  7,  gives  —  5. 
Dividing  —  5  by  2,  the  quotient  is  fractional ;  therefore 
2  is  not  a  root. 

l-l-7+l-f-G|_3  1_1_7+14-G  |_-2 

--1   -2       1       2  -13       1-3 

~~0  -3  -6       3  ~0       2  -6  -2 

In  these  cases,  each  quotient  is  integral,  and  the  last 
quotient  added  to  unity  gives  0;  therefore  3  and  —2  are 
roots. 


SOLUTION    OF   HIGHER   EQUATIONS.  503 

Thus,  tlie  four  roots  of  the  given  equation  are  1,  —1,  3, 
and  -  2. 

Note.  There  is  no  necessity  for  trying  +  6,  in  the  above  example, 
for  we  know  that  the  equation  cannot  have  more  than  two  positive 
roots. 

EXAMPLES. 

692.  Find  all  the  commensurable  roots  of  each  of  the 
following  equations,  and  the  remaining  roots  when  possible 
by  methods  already  given: 

1.  a;^-8x-2  +  19i(;-12  =  0.         4.    2a.-3  +  a;2_  23x  +  20  =  0. 

2.  a;3-31a;-30  =  0.  5.    x" -7x- ~Ux +  'i8  =  0. 

3.  x''  +  5x'-6x-24:  =  0.  6.    3a;3  +  2a;- -  3x  -  2  =  0. 

7.  a;''  +  2.r^-7cc--8a:  +  12  =  0. 

8.  x'  +  6x^  +  x--2ix-20  =  0. 

9.  4:X*-12x'  +  Sx^-\-13x~6  =  0. 

10.  x*  +  llx^-\-4=lx''-\-61x  +  30  =  0. 

11 .  x*  +  x^  -  31  a--  +  71  x  -  42  =  0. 

12.  4=x*  -  31  x'  +  21  a;  -f  18  =  0. 

13.  a;*-lla;«  +  35a;--13a;-60  =  0. 

14.  x^  +  14 af^  -  6  .T-  +  45  x  -  54  =  0. 

15.  9x*-16x^-3x  +  4:  =  0. 

16.  a;''-7x'^  +  15x2-a;-24  =  0. 

RECIPROCAL   OR  RECURRING   EQUATIONS. 

693.  A  Reciprocal  Equation  is  one  such  that  if  any  quan- 
tity is  a  root  of  the  equation,  its  reciprocal  is  also  a  root. 

It  follows  from  the  above  that,  if  -  is  substituted  for  x 

X 

in  a  reciprocal  equation,  the  transformed  equation  will  have 
the  same  roots  as  the  given  equation. 


504  COLLEGE   ALGEBRA. 

694.    Let 

be  a  reciprocal  equation. 

Putting  -  in  place  of  x,  the  equation  becomes 

X 

aj"      x"~^      x"-^  ^         X        '■  ' 

or,  by  clearing  of  fractions,  and  reversing  the  order  of  the 

terms, 

_?)„a;"  +l)„-ix"-^  +P„-2-'c""^  ^ VV-i^  -Vlh^  +Po  =  0.    (2) 

By  Art.  693,  this  equation  has  the  same  roots  as   (1), 

an&  hence  the  following  relations  must  hold  between  the 

coefficients  of  (1)  and  (2): 

Po  =  ±  P„,  i\  =  ±  i>,,  ^1,  P2  =  ±  Pn-1^  etc. ; 

or,  in  general,  Pr  =  ±  Pn-r ; 

all  the  upper  signs,   or  all  the  lower  signs,  being  taken 

together. 

We  may  then  have  four  varieties  of  reciprocal  equations  : 

1.  Degree  odd,  and  coefficients  of  terms  equally  distant 
from  the  extremes  of  the  first  member  equal  in  absolute 
value  and  of  like  sign ;  as,  a^  —  2a;^  —  2  a;  +  1  =  0. 

2.  Degree  odd,  and  coefficients  of  terms  equally  distant 
from  the  extremes  of  the  first  member  equal  in  absolute  value 
and  of  opposite  sign ;  as,  3  a^  +  2  cc*  —  cc'^  +  x-  —  2  cc  —  3  =  0. 

3.  Degree  even,  and  coefficients  of  terms  equally  distant 
from  the  extremes  of  the  first  member  equal  in  absolute 
value  and  of  like  sign ;  as,  x^  —  5  x^  +  6  a;^  —  5  a;  +  1  =  0. 

4.  Degree  even,  and  coefficients  of  terms  equally  distant 
from  the  extremes  of  the  first  member  equal  in  absolute 
value  and  of  opposite  sign,  and  middle  term  wanting;  as, 
2x''  +  3a;'^  -  7.1;^  +  7a;^  -  3a;  -  2  =  0. 

On  account  of  the  pro]ierties  stated  above,  reciprocal 
('r|uations  are  also  ml  led  RRcurring  Efjtiations. 


SOLUTION    OF   HIGHER   EQUATIONS.  505 

695.  Every  reciprocal  equation  of  the  first  variety  may 
be  written  in  the  form 

Po^^"  +Pl.^*""^  +P2^""^  H \-lh^'+PiX  +  Po  =  0, 

or,     i5o(^"  +  t)  +i^i^(x''-'  +  l)  +ij,x\x"  '  +  l)+-  =  0 ; .  (1) 
the  number  of  terms  being  even. 

By  Art.   113,   since  n  is  odd,  each  of   the  expressions 
x"  +  1,  X"  -  +  1,  etc.,  is  divisible  by  a;  +  1. 
Therefore  —  1  is  a  root  of  the  equation. 
Dividing  the  first  member  of  (1)  by  x  + 1,  the  depressed 
equation  is 

Po(^" ' '  —  cc"  -  +  a" '^ +x-  —  x  -{-1) 

+  p,x  (x"-^  -  a;"-^  4-  .^■""^ {-x'  —  x  +  l) 

-\-  2h^-{x''-^  —  cc"-"  +  x"^''  —  •  •  •  +  a;-  —  X  +  1)  +  •  •  •  =  0, 

or,     po^^""'  +  {Pi  -i^o)  a;"--  +  (p.  -Pi  -i-Po)  x""-'^  +  ••• 
+  ( Vi  -  Pi  +  i^o)  X-  +  (Pi-  ih)  x  +  po  =  0 ; 

which  is  a  reciprocal  equation  of  the  third  variety. 

696.  Every  reciprocal  equation  of  the  second  va,riety 
may  be  written  in  the  form 

l^oX"  +2hx''-'  +2hx''-^  -\ (•••  +P2X^  +PiX  +po)  =  0, 

or,     2,^{x"-l)+2hx{x^-'-l)+P2x\x'^-*-l)-\---'=0.     (1) 

Since  each  of  the  expressions  x"  —  1,  x"''-  —  1,  etc.,  is 
divisible  by  ic  —  1,  +  1  is  a  root  of  the  equation. 

Dividing  the  first  member  of  (1)  by  a;  —  1,  the  depressed 
equation  is 

PoX''^^  +  {Pi  +  Po)  x"--  +  (P2+  Pi  +Po)  x"-^  +  ... 

+  {Pi  +Pi  +Po)  X-  +  (p,  +Po)  X  +po  =  0; 

which  is  a  reciprocal  equation  of  the  third  variety. 


506  COLLEGE   ALGEBRA. 

697.  Every  reciprocal  equation  of  the  fourth  variety  may 
be  written  in  the  form 

PoX^+PiX''-^+2hx"^^-\ (■'•+P2X^+PyX+Po)  =  0, 

or,    Po{x--l)+PiX{x'^-'-l)+p,x''(x--'-l)-\-...  =  0;    (1) 
the  number  of  terms  being  even  (Art.  694). 

Since  each  of  the   expressions  x"  —  1,  x"~^  —  1,  etc.,    is 
divisible  by  x^—  1,  both  1  and  —  1  are  roots  of  the  equation. 

Dividing  the  first  member  of  (1)  by  x^  —  1,  the  depressed 
equation  is 

Po{x'^~'  +  x"-"  +  ...  -f  a;*  +  a;-  +  1) 

+PiX  {x--*  +  x--'  +  ...  +x'  +  x'  +  1) 

+P2x'{x-  '  +  x--'  Jr-+x'  +  x'  +  1)  -^  ...  =  0, 

or,  Pq a;"-2  +  j^i  x"'^  +  (p.  +  Po)  ^"  "  +  •  •  • 

+  (P2  +Po)  ^"  +  Pi  a^  +  Po  =  0 ; 

which  is  a  reciprocal  equation  of  the  third  variety, 

698.  Every  reciprocal  equation  of  the  third  variety  may  be 
reduced  to  an  equation  of  half  its  degree. 

Let  the  equation  be 

Poa^^+Pia^""^-! +i'm^"'H +Pia;+po  =  0. 

Dividing  through  by  x"  the  equation  may  be  written 


Put 

.+1  = 

X 

=  y- 

Then, 

--1- 

-h 

^J- 

2  =  2/— 2; 

--1- 

=(- 

^)e 

'^^-i: 

4) 

=  ui>/' 

-2)- 

-y  =  f-^ 

?/; 

SOLUTION   OF   HIGHER   EQUATIONS.  507 

The  general  law  is  expressed  by 

an  expression  of  the  rth  degree  with  respect  to  y. 

Substituting  these  values  in  (1),  the  equation  becomes 

go  2/"*  +  (iiv"'^'^  +  Q^y'"'-  +  •••  =  0. 

699.  It  follows  from  Arts.  695  to  698  that  any  reciprocal 
equation  of  the  degree  2  m  +  1,  and  any  reciprocal  equation 
of  the  fourth  variety  of  the  degree  2  m  +  2,  can  always  be 
reduced  to  an  equation  of  the  mth  degree. 

700.  1.  Solve  2.x'^- 5a;*- 13ar'  +  13iK2  +  5x- 2  =  0. 
The  equation  being  of  the  second  variety,  one  root  is  1 

(Art.  696). 
Dividing  by  a;  —  1,  the  depressed  equation  is 

2a;*  -  3x3  -  16x- -  3x  +  2  =  0 ; 

a, reciprocal  equation  of  the  third  variety. 

Dividing  by  x",  2  fx'  -\-  ^}j  -  sfx  +  -\  -  16  =  0. 

Putting  x-\--  =  y,  and  x^  +  —  =  ?/2  _  2  (Art.  698),  we  have 

2(/-2)-3^-16  =  0. 
Solving  this  equation,  y  =  4  or  —  -• 

Taking  the  first  value,        x  +  -  =  4,  ox  x-  —  4x  =  —  1. 

X 

Whence,  x-  =  2  ±  Vo. 


608  COLLEGE   ALGEBRA. 

1         5 
Taking  the  second  value,  cc  +  -  = ,  ot  2x^  +  5x  =  —  2. 

X         2 

Whence,  a;  =  —  2  or 

2 

Thus  the  roots  of  the  given  equation  are  1,   —  2,   —  -, 
and  2  ±  V3.  ^' 

Note.    That  2  +  VS  and  2  —  V3  are  reciprocals  may  be  shown  by 
multiplymg  them  together  ;  thus,  (2  +  VS)  (2  —  VS)  =  4  —  3  =  1. 


EXAMPLES. 

Solve  the  following  equations  : 
2.    C)x^-7x''-7x-\-6  =  0.       4.    5  a;^ -|- 26  .^•3  -  26  .t  -  5  =  0. 
•3.    x^  +  ox^  —  ox  —  l=0.         5.    ar*  —  aar  +  «x  —  1  =  0. 

6.  45a;^-48ar'-250ar'-48a;  +  45  =  0. 

7.  .^'^-29a,•3  +  29a;^-l  =  0. 

8.  x^  +  7x*  +  x'+x^  +  7x  +  l  =  0. 

9.  24  a.-^  -  34 a;*  -  67  ar^  +  67  a^  + 34  X- 24  =  0. 

10.  3ar'  +  16a;''  +  29ar''  +  29a;2+16a;  +  3  =  0. 

11.  4a;«-29a:^  +  55a;''-55.'c2  +  29a;-4  =  0. 

701.  Binomial  Equations. 

A  Binomial  Equation  is  an  equation  of  the  form  a;"  =  a. 

Binomial  equations  are  also  reciprocal  equations,  and,  in 
certain  cases,  may  be  solved  by  the  method  of  the  preced- 
ing article. 

702.  Putting  a;  =  ay,  the  equation  x"  =  ±  a"  becomes 
2/"  =  ±  1 ;  Avhich  is  a  form  to  which  every  binomial  equa- 
tion may  be  reduced. 

In  Arts.  351  and  358,  methods  were  given  for  the  solution 
of  the  V)in()iuial  equations  x^  =  ±  1,  a;*  =  ±  1,  and  a:"  =  ±  1. 


SOLUTION   OF   HIGHER   EQUATIONS.  509 

The  forms  a^  =  ±  1  are  readily  solved  by  the  method  of 
Art.  700. 

Binomial  equations  of  any  degree  may  be  solved  by  a 
method  involving  Trigonometry, 

EXAMPLES. 
Solve  the  following  equations  : 
1.    x'  =  l.         2.    af  =  -l.         3.03^  =  32.     (Putx  =  2^.) 

703.  The  Cube  Roots  of  TJnity. 

By  Ex.  3,  Art.  351,  the  roots  of  the  equation  a;''  =  1  are 

1,    -1±VE?,    and    -^-^^. 

2  '  2 

The  third  root  is  the  square  of  the  second ;  for 


14.V_3Y_  1-2V-3-3  ^  -l-V-3 

4  2 


(-^¥-^J 


Hence,  if  the  second  root  is  denoted  by  w,  the  three  cube 
roots  of  unity  are  1,  w,  and  or. 

3  1 

Or,  since  w^  =  1,  they  are  1,  w,  and  —  or  -• 

(O  to 

704.  If  the  second  root  is  denoted  by  a,  the  three  roots 
are  a,  aw,  and  aw- ;  for  these  are  respectively  equal  to  w,  w^, 
and  co''  or  1. 

In  like  manner,  if  the  third  root  is  denoted  by  a,  the 
three  roots  are  a,  aw,  and  awl 

Hence,  if  either  of  the  cube  roots  of  a  quantity  is  denoted 
by  a,  the  other  two  roots  are  aw  and  aw-. 

CUBIC   EQUATIONS, 

705.  A  Cubic  Equation  is  an  equation  of  the  third  degree 
(Art.  179),  containing  but  one  unknown  quantity. 


510  COLLEGE   ALGEBRA. 

706.  By  Art.  652,  the  cubic  equation 

x^  +  pi  X-  +  P2  ^  -\-  Pz  =  0) 
where  pi  is    not   zero,   may  be   transformed   into  another 
whose  second  term  shall  be  wanting,  by  substituting  y  —■— 
in  place  of  x. 

Therefore,  every  cubic  equation  caii  be  reduced  to  the  form 

ar*  +  rtx  +  &  =  0. 

707.  Cardan's  Method  for  the  Solution  of  Cubics. 

Let  it  be  required  to  solve  the  equation  x'  +  aa;  +  &  =  0. 
Putting  x  =  y  +  z,  the  equation  becomes 

f  +  oyz{y  +  2)  +  r  +  a(2/  +  2)  +  &  =  0, 

or,  f  +  z'  +  {^yz-\-a){y  +  z)  +  h  =  0. 

We  may  take  y  and  z  in  such  a  way  that  3^2  +  a  shall  be 
equal  to  zero ;  whence, 

Then,  2/^  +  2^  +  5  =  0.  (2) 

Substituting  in  this  the  value  of  z  from  (1),  we  have 
3 _ _a^  +  &  =  0,  or  y^J^hf=^—- 

This  is  an  equation  in  the  quadratic  form  (Art.  3G3) ; 
solving  by  the  rules  for  quadratics,  we  have 


Then  by  (2),  ^  =  -  f  -  h  = -^^-^  yj^^-  (4) 

Since  x  =  y  +  z,  the  values  of  x  corresponding  to  the  upper 
and  lower  signs  in  (3)  and  (4)  will  evidently  be  the  same. 


SOLUTION   OF    HIGHER   EQUATIONS.  511 

Therefore, 

^•=xI(-IWM)s'(-2-nR>  ^^> 

The  remaining  roots  may  be  found  by  depressing  the  equa- 
tion (Art.  635),  or  by  the  method  explained  in  Art.  710. 

708.  From  (1),  Art.  707,  we  derive  the  following  rule: 
To  solve  a  cubic   equation   of  the  form   x^  -\-ax  +  b  =  0, 

substitute  y fo7-  x. 

709.  1 .  Solve  the  equation  ar'  +  3  a;-  —  6  x  +  20  =  0. 

We  first  transform  the  equation  into  another  whose 
second  term  shall  be  wanting ;  putting  x  =  y  —  1  (Art.  706), 
we  have 

f  -  3y2  +  3y  -  1  ^  3f'  -  Gy  +  3  -  6y  -[-  6  -\-  20  =  0, 

or,  y'-9y  +  2S  =  0. 

The   latter   equation   may   now   be    solved    by   putting 
3 
y  =  z  +  -  (Art.  708);  or,  by  substituting  a  =  —  9  and  6  =  28 

in  (5),  Art.  707. 

Using  the  second  method,  we  have 

y  =  ^/_U-hVi96-27  +  ^-14-Vl96-27 

=  ^^ri+i/Zr27  =  -l-3  =  -A. 

Therefore,  x  =  y  —  1  =  —  5. 

Dividing  the  first  member  of  the  given  equation  by  x  -f-  5, 
the  depressed  equation  is 

x--2x  +  4:  =  0. 

Solving,  we  have     x  =1  ±  V—  3. 

Thus    the   roots   of    the    given   equation    are    —  5   and 


512  COLLEGE    ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations  : 

2.  X-3  + 15a; +  124  =  0.  7.  ar'^ -f- a;^  -  33  a;  +  63  =  0. 

3.  ar^- 27a;- 54  =  0.  8.  a;'^  +  12a;-'  + 57a;  + 74  =  0, 

4.  a;'5  +  105a;-218  =  0.  9.  ar^  -  4x2- 11a;- 6  =  0. 

5.  ar^_ 6x2- 33a; -70=0.     ^q.  .r^- 2.t2  +  3  =  0. 

6.  a;3-9a;2  + 63a; +  73  =  0.     11.  .^3  + a.-^- 7a;  -  52  =  0. 

12.    Find  one  root  of  .r''  —  .x  +  1  =  0. 

Note.  A  cubic  equation  having  a  commensurable  root  is  solved 
more  easily  by  the  method  of  Art.  689  than  by  Cardan's  rule. 


710.    If  h  is  any  one  of  the  cube  roots  of \-  a/—  +  — , 

and  Ti  any  one  of  the  cube  roots  of \\ — |- — ,  the 

^  2       \4      27 

three  cube  roots  of  the  first  expression  are  h,  hw,  and  hw,% 
and  the  three  cube  roots  of  the  second  are  k,  kw,  and  kw^ 
(Art.  704). 

This  would  apparently  indicate  that  x  has  nine  different 
values. 

But  by   (1),  Art.  707,  yz  =  -  -  ;  that  is,  the  product  of 
o 


Hence  the  only  possible  values  of  x  are 

h  +  k,  ho)  +  kw^,  and  har  +  yfcw ; 

for  in  each  of  these  the  product  of  the  terms  is  hk ;  that  is, 

3  IP      P      oF  a 

\ ,  or  —    ;  while  in  any  other  case  the  product 

\4       4      27  3'  -^  ^ 

is  either  —  -la  or  —  -  w. 
3  o 


SOLUTION   OF   HIGHER   EQUATIONS.  513 

Putting  for  w  and  w'  their  values  (Art.  703),  tlie  second 
and  third  values  of  x  become 


and  hl~'^~y~''^]+k 


1^-). 


Hence  the  three  values  of  x  are 
li  +  k,  _^r_  +  ___V-3,  and ^ Y^^~^- 

Thus  in  Ex.  1,  Art.  709,  7i  =  -  1  and  k  =  -  3. 

Then  the  values  of  y  are  -  4,  2  +  V^^,  and  2  -  V^3. 

EXAMPLES. 

Solve  the  following  equations : 

1.    a^  +  6a;  +  2  =  0.  2.    af'  +  9rc-6  =  0. 

711.  Discussion  of  the  Roots. 


«-^(-2W!-l)-"=^'(-|-VM/ 

the  roots  of  af^  +  aa;  -f-  6  =  0  are 

fe  +  fc,  and  -^*±'l^V^^  (Art.  710). 

2  ^ 

1.  If  a  is  positive,  or  if  a  is  ne-;ative  and  —  numerically 
i.'ss  than  — ,  /t  and  k  are  real  and  unequal. 

Therefore  one  root  is  real,  and  the  other  two  imaginary. 

2.  If  a  is  negative,  and  ^  numerically  equal  to  p  li  and 
fc  are  real  and  equal,  and  Ji  —  k  is  zero. 

Hence  the  roots  are  all  real,  and  two  of  them  are  equal. 


514  COI.LEGE  ALGEBRA. 

3.    If  a  is  negative,  and  —  numerically  greater  than   — , 

Ji  i  4 

the  values  of  h  and  A;  involve  imaginary  expressions. 

In  this   case,  h  must   have   some  value    of    the    form 
h'-\-hW^-i,  where  A'  a^d  k'  are  real  (Art.  627) ;  that  is, 


Raising  both  members  to  the  third  power, 


Equating  the  real  and  imaginary  parts  (Art.  338), 
^.  =  h''-3h'k'\ 


9 


and  J^  +  1^  =  oh'-k'V^l  -  k'W- 1. 


4      27 
Subtracting, 


_  ^  _^1'  +  |-  =  /i'^-37i'2fc' V^-  3/i'A;'2+  fc'^V^. 

Extracting  the  cube  root, 

k  =  h'-k'V^l.  (2) 

From  (1)  and  (2), 

h  +  k  =  2h',  and  h-k  =  2 k'V^^. 
Then  the  three  roots  are  27i'  and  —  h'±  k'V—  iV—  3. 
That  is,  27i'  and  -h'^  k'VS. 
Therefore  the  roots  are  real  and  unequal. 

In  the  above  case,  Cardan's  method  is  of  no  practical 
value ;  for  since  there  is  no  method  in  Algebra  for  finding 
the  cube  root  of  an  expression  which  is  in  .the  form  of  a 
rational  expression  plus  a  quadratic  surd  (Art.  291),  the 
values  of  h  and  k  cannot  be  found.  In  this  case,  which  is 
called  the  Irreducible  Case,  Cardan's  method  is  said  to  fail. 


SOLUTION    OF   HIGHER   EQUATIONS.  515 

It  is  possible,  in  cases  wliere  Cardan's  method  fails,  to  find 
the  roots  by  Trigonometry  ;  but  in  practice  it  is  easier  to  lind 
them  by  Art.  689,  or  by  Horner's  method  (Art,  718),  accord- 
ing as  the  equation  has  or  has  not  a  commensurable  root. 

712.  Consider  the  equation  x'  +  ax-  +  bx  +  c  =  0. 

Putting  x  =  y  —  -,  the  equation  becomes 
o 

,  0  ,  a-y      a^   ,       .-,      2a-y  ,  a*  ,  ,        ab  ,  ^      ^ 

Multiplying  the   roots   by    3    (Art.  644),   the   equation 

becomes 

f  +  3{3b~a-)y-\-2cc'-9ab-{-27c  =  0. 

Then  it  follows  from  Art.  711  that  * 

1.  If  3b  — a^  is  positive,  or  if  36  — cr  is  negative  and 
4(3&  —  a2) 3  numerically  less  than  {2a^ —  9  ab -{-27  c)-,  the 
given  cubic  has  one  real  and  two  imaginary  roots. 

2.  If  3&  —  a^  is  negative,  and  4(3&  —  a-y  is  numerically 
equal  to  {2a^  —  9ab  +  27c)^,  the  roots  are  all  real,  and  two 
of  them  are  equal. 

3.  If  3b  — a-  is  negative  and  4(3&  — a-)'  numerically 
greater  than  (2a^  —  9ab  +  27c)-,  the  roots  are  all  real  and 
itnequal. 

BIQUADRATIC   EQUATIONS. 

713.  A  Biquadratic  Equation  is  an  equation  of  the  fourth 
degree  (Art.  179),  containing  but  one  unknoAvn  quantity. 

714.  Euler's  Method  for  the  Solution  of  Biquadratics. 

By  Art.  652,  every  biquadratic  equation  can  be  reduced 
to  the  form 

x^+ox-  +  b.v-\-c  =  0.  (1) 


516  COLLEGE   ALGEBKA. 

Let  xi=u  +  y-\-z;  then, 

x'==ir  +  t/  +  z'  +  2uy  +  2yz  +  2zu, 
or,  x'  -  (u-  +  y'  +  2')  =  2  {uy  +  yz  +  zu). 

Squaring  both  members,  we  have 
x^  —  2ar{v?  +  y-+  z-)  -\-  (u^  +  2/^  +  zy=  4:(uy  +  yz  -{-  zuY 

=  4(rty  +  y-z'^  +  zhi?)  +  8  uyz{u  +  y+z). 
Substitiiting  x  for  it.+  ?/  +  ^,  and  transposing, 
x'^  —  2af{u^  +  ?/2  +  2;-)  —  8  uyzx 

+  (m2+  ?/2+  2-)2  -  4(My  +  2/V  +  zhv")  =  0. 

This  equation  will  be  identical  with  (1)  provided 

a  =  -2{it'  +  y'  +  z'), 

b=-8mjz,  (2) 

and      c  =  ( u-  +  r  +*") "  -  H^y-  +  y-z^  +  2^%^) . 
Therefore,  w"  +  2/'  +  2'  =  ~  «> 

"^         64' 

and  uY~  +  2/^.^  +  .V  =  (lL±l±fr^ 

4  a  -  —  4c 


4  16 

If,  now,  we  form  the  cubic  equation 
^3  _  (,^2  _|.  2/2  +  z-)t-  +  (?ty  +  y-z-  +  ^-u2)«  -  ir^/V  =  0, 
the  values  of  t  will  be  u%  /,  and  z-  (Art.  637). 
Hence,  if  the  roots  of  the  cubic  equation 

are  I,  m,  and  n,  -we  shall  have 

u=±  Vl  >l  ^  ±  Vm,  and  z  =  ±  Vn. 


SOLUTION   OF   HIGHER   EQUATIONS.  517 

NoAv  x=:  u  +  y  -\-z;  and  since  each  of  the  quantities  u,  y, 
and  z  has  two  vakies,  apparently  x  has  eight  vakies. 

But  by  (2),  the  product  of  the  three  terms  whose  sum  is 

a  vakie  of  x  must  be  equal  to  —  -• 

o 

Hence  the  only  values  of  x  are,  when  h  is  positive, 

—  V^  —  Vm  —  V/*,   —  V^  +  V«t  4-  V?i, 
V^  —  Vm  +  Vrt,  and  VZ -f  V»i  —  Vh; 

and  when  h  is  negative, 

V^  +  Vm  +  Vn,   vT  —  Vwi  —  V^i, 

—  V^  +  V»i  —  V»,  and   —  v'/  —  V»i  +  V». 
Equation  (3)  is  called  the  auxiliary  cubic  of  (1). 

715.   1.  Solve  the  equation  x^  -  46  aj-  -  24  x  +  21  =  0. 
Here,  a  =  —  46,  6  =  —  24,  and  c  =  21 ;  whence, 

^^'-^^=127,  and  ^  =  9. 
16  64 

Then  the  auxiliary  cubic  is  f  —  23 «-  +  127 « -  9  =  0. 
By  the  method  of  Art.. 689,  one  value  of  t  is  9. 
Dividing  the  first  member  by  ^  —  9,  the  depressed  equii/- 
tion  is  «-  — 14^  +  1  =0. 

Solving,  we  obtain  t  =  l  ±  V49  —  1  =  7  ±  4  V3. 
Proceeding  as  in  Art.  313,  we  have 

V(7  ±  4  V3)  =  V(4  ±  2  Vl2  +  3)  =  2  ±  V3. 
Then  since  h  is  negative,  the  four  values  of  x  are 
3  +  2+V3  +  2-V3,  3-2-V3-2+V3, 
-3  +  2+V3-2+V3,  and  _  3  -  2- V3  +  2- Va 
That  is,  7,  —  1,  -  3  +  2  V3,  and  -  3  -  2  V3. 


518  COLLEGE   ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations  : 

2.  a;4-42a;2  +  64aj+105  =  0. 

3.  a;"  -  54 tc-- 24  a; +  77=0. 

4.  a;^-76a^- 16a; +  896  =  0. 

5.  .r^-10a;2  + 20.x -16  =  0. 

6.  x'-  36 x^  +  16  X-  +  195  =  0. 

7.  a-" +  4x3  + 3a;- -44a; -84  =  0. 

716.   Discussion  of  the  Roots. 

The  auxiliary  cubic  of  x^  +  ax-  +  &x  +  c  =  0  is 

^3      a^2      o^^-4c        6^^^  (Art.  714). 
2  16  64         ^  ^ 

Since  the  last  term  is  essentially  negative,  the  equation 
must  have  either  three  positive,  one  positive  and  two  nega- 
tive, or  one  positive  and  two  imaginary  roots  (Art.  637). 

Multiplying  the  roots  by  4  (Art.  644),  the  equation 
becomes 

e-\-2at-  +  {a--4:c)t-b^  =  0. 

Denoting  2a,  a^  — 4c,  and  —  h^  by  a',  h',  mid  c',  we  have 
36'-  a'-  =  3(a-  -  4c)  -  4a- =  -  (a-  +  12c), 
and         2a"^-  9a'b'+  27c'=  16  a"  -  18  a (a^  -  4c)  -  276- 
=  - (2  a'^  -  72 ac  + 27  6-). 
Then  it  follows  from  Art.  712  that : 

1.  If  a^  +  12c  is  negative,  or  if  a^  +  12c  is  positive  and 
4(a-  +  12c)'^  less  than  (2a»- 72 ac +276^)2,  the  auxiliary 
cubic  has  one  positive  and  two  imaginary  roots. 

If  V^  =  p,  Vm  =  g  +  rV—  1,  and  Vh  =  Q  —  rV—  1,  the 
roots  of  the  biquadratic  are 
—  p  ±  2q  and  p  ±  2r  V—  1,  or  p±2q  and  —  p  ±2rV  — 1, 
according  as  b  is  positive  or  negative. 


SOLUTION   OF   HIGHER   EQUATIONS.  51? 

That  is,  the  biquadratic  has  two  real  and  two  imagioary 
roots. 

2.  If  a-  +  12c  is  positive,  and  ^{0^  -\-12cy  is  equal  to 
(2a^  —  72  ac  +  27  b^y,  the  cubic  has  two  roots  equal. 

If  Vw  =  V»i,  the  roots  of  the  biquadratic  are 
—  Vl  ±2  Vm,  VT,  and  V^  or  ■\/T±  2  Vm,  —  VT,  and  —  V^ 
according  as  b  is  positive  or  negative. 

That  is,  the  biquadratic  has  two  roots  equal. 

3.  If  a^+12c  is  positive  and  4(a^  + 12  c)'^  greater  than 
(2a^—  72  ac-\- 27  b^y,  the  cubic  has  either  three  positive,  or 
one  positive  and  two  negative  roots. 

In  the  first  case,  the  roots  of  the  biquadratic  are  all  real; 
in  the  second  case,  they  are  all  imaginary. 

4.  If  a2+12c=0  and  2a^-72ac+27b''=0,  then  c  =  -^. 
Substituting  from  the  third  equation  in  the  second, 

8  a^  +  27b^  =  0,  or  a  =  -  J  6^  ;  whence,  a--4c  =  —  =  3  b\ 


In  this  case,  the  auxiliary  cubic  becomes 

16  64       '        V       4y 

b^ 


4 


and  each  of  its  roots  is  equal  to 

The  roots  of  the  biquadratic  are  —- — ,  — ,  — ,  and  — ,  or 

2      2     2  2 

3&^        b^        b^         1       b^^  T  ,    . 

, , ,  and  —  — ,   according    as    0   is   positive   or 

negative ;  that  is,  the  biquadratic  has  three  roots  equal. 

5.    If  a^  —  4c  =  0  and  &  =  0,  the  biquadratic  becomes 

x'^  +  ax-  +  ^'  =  0,  or  (x"  +  "Y  =  0, 

and  its  roots  are  ±-v/—  -  and  ±  a/— -• 

That  is,  the  biquadratic  has  two  pairs  of  equal  roots. 


520  COELEGE  ALGEBRA. 


INCOMMENSURABLE   ROOTS. 

717.  We  will  now  show  how  to  find  the  approximate 
numerical  values  of  those  roots  of  an  equation  which  are 
not  commensurable  (Art.  689). 

718.  Horner's  Method  of  Approximation.  '  'w-  u  q, 

Let  it  be  required  to  find  the  approximate  value  of  the 
root  between  3  and  4  of  the  equation  {^^-^  }^  - 

ar^_3x'2-2a;-f  5  =  0. 

Diminishing  the  roots  of  the  given  equation  by  3,  by  the 
method  explained  in  Art.  650,  we  have 

1     _3     -2     +5  [3_ 
3         0-6 
0     -2     -1 
3         9 
3         7 
3 
6 
The  transformed  equation  is  ^/'^  +  6?/^  +  7?/  —  1  =  0.      (1) 
This  equation  is  known  to  have  a  root  between  0  and  1 ; 
if,  then,  we  neglect  the  terms  involving  ?/^   and  y^,  we  may 
obtain  an  approximate  value  of  y  by  solving  the  equation 
7?/  —  1  =  0 ;  thus,  approximately,  y  =  .\  and  x  =  3.1. 
Diminishing  the  roots  of  (1)  by  .1,  we  have 


■6    +7 
.1       .61 

-1      L 

.761 

6.1     7.61 
.1       .02 

-    .239 

6.2     8.23 
.1 

6.3 

The  transformed  equation  is  z''  +  6.3 2^  +  8.23 z  —  .239  =  0. 


SOLUTION   OF   HIGHER   EQUATIONS.  521 

Neglecting  the  z"  and  7}  terms,  we  have,  approximately, 

8.23 

Thus  the  value  of  x  to  two  places  of  decimals  is  3.12. 

The  process  may  be  continued  until  the  value  of  the  root 
has  been  found  to  any  desired  degree  of  precision. 

The  work  is  usually  arranged  in  the  following  form,  the 
coefficients  of  the  successive  transformed  equations  being 
denoted  by  (1),  (2),  (3),  etc.: 


-3 

_ 

-2 

+  5              1  ai28 

3 

0 

-6 

0 

- 

-2 

(1) 

-1 

3 

_9 

.761 

3 

(1)' 

7 

(2) 

-    .239 

3 

,61 

.167128 

(1)    6 

7.61 

(-) 

-    .071872     . 

.1 

.62 

6.1 

(2) 

8.23 

.1 

.1264 

6.2 

8.3564 

.1 

.126<^ 

(2)    6.3 

(-/ 

8.4832 

.02 

6.32 

.02 

6.34 

.02 


(3)    6.36 

Dividing  .071872  by  8.4832,  we  have  .008  suggested  as 
the  fourth  figure  of  the  root. 

Thus  the  value  of  x  to  three  places  of  .decimals  is  3.128. 


522  COLLEGE   ALGEBRA. 

We  derive  from  the  above  the  following  rule  for, finding 
the  approximate  value  of  a  positive  incommensurable  root : 

Find  hy  Arts.  G71,  073,  or  687,  or  by  Stur7n^s  Theorem,  the 
integral  part  of  the  root.     (Compare  Art.  674.) 

Transform  the  given  equation  into  another  ivhose  roots  shall 
be  less  by  this  integral  part. 

Divide  the  absolute  value  of  the  last  term  of  the  transformed 
equation  hy  the  absolute  vahie  of  the  coefficient  of  the  first 
poiver  of  the  unJcnoivn  quantity,  and  ivrite  the  approximate 
value  of  the  result  as  the  next  figure  of  the  root. 

Transform  the  last  equation  into  another  ichose  roots  shall 
be  less  by  the  figure  of  the  root  last  obtained,  and  divide  as 
before  for  the  next  figure  of  the  root;  and  so  on. 

Note.  In  practice,  the  work  may  be  contracted  by  dropping  sucli 
decimal  figures  from  the  right  of  each  column  as  are  not  needed  for  the 
required  degree  of  accuracy. 

719.  To  find  the  approximate  value  of  a  negative  incom- 
mensurable root,  change  the  signs  of  the  alternate  terms  of 
the  equation  beginning  with  the  second  (Art.  643),  and  find 
the  corresponding  positive  incommensurable  root  of  the 
transformed  equation. 

The  result  with  its  sign  changed  will  be  the  required 
negative  root. 

720.  In  finding  any  particular  root-figure  by  the  method  . 
exi)lained  in  Art.  718,  we  are  liable,  especially  in  the  first 
part  of  the  process,  to  get  too  great  a  result ;   the  same 
thing  occasionally  happens  when  extracting  square  or  cube 
roots  of  numbers. 

Such  an  error  may  be  discovered  by  observing  the  signs 
of  the  last  two  terms  of  the  next  transformed  equation ;  for 
since  each  root-figure  obtained  as  in  Art.  718  must  be  posi- 
tive, the  last  two  terms  of  the  transformed  equation  must 
be  of  opposite  sign. 


SOLUTION  OF   HIGHER   EQUATIONS.  52^ 

If  this  is  not  the  case,  the  last  root-figure  must  be 
diminished  ui*fcil  a  result  is  obtained  which  satisfies  this 
condition. 

Let  it  be  required,  for  example,  to  find  the  root  between 
0  and  —  1  of  the  equation  or^  -f  4  a;-  —  9  a;  —  5  =  0. 

Changing  the  signs  of  the  alternate  terms  beginning  with 
the  second  (Art.  719),  we  have  to  find  the  root  between 
0  and  1  of  the  equation  a^  —  4a^  —  9cc  +  5  =  0. 

Dividing  5  by  9,  we  have  .5  suggested  as  the  first  root- 
figure  ;  but  it  will  be  found  that  in  this  case  the  last  two 
terms  of  the  second  transformed  equation  are  — 12.25  and 
—  .375. 

This  shows  that  .5  is  too  great ;  we  then  try  .4,  and  find 
that  the  last  two  terms  of  the  second  transformed  equation 
are  of  opjjosite  sign. 

The  work  of  finding  the  first  three  root-figures  is  shown 
below : 

.469 


1 

-4 
.4 

-  9 

-  1.44 

+  5 
-4.176 

-3.6 
.4 

-  10.44 

-  1.28 

(1) 

.824 
-  .713064 

(1) 

-3.2 
.4 

-2.8 
.06 

(1) 

-11.72 

-  .1644 
- 11.8844 

-  .1608 

(2) 

.110936 

-2.74 
.06 

(2) 

- 12.0452 

-2.68 
.06 

(2) 

-2.62 

The  required  root  is  therefore  —.469,  to  three  i^laces  oE 
decimals. 


o24  COLLEGE   ALGEBRA. 

In  any  case,  the  root-figure  to  be  taken  is  the  greatest 
number  luhich  will  ensure  that  the  last  two  terms  of  the  next 
transformed  equation  shall  be  of  opposite  sign. 

Note.  In  some  cases,  the  first  trausforined  equation  gives  very 
little  information  in  regard  to  the  first  decimal  root-figure. 

Thus,  in  the  equation  x*  — 7a;2  — 5  a;  — 1  =0,  which  has  a  root  be- 
tween 2  and  3,  the  first  transformed  equation  is 

yi  +  8?/3  +  17y2_y_23  =  0, 

the  last  two  terms  being  negative. 

The  rule  directs  us  to  take  the  greatest  number,  less  than  unity, 
which  will  ensure  that  the  last  two  terms  of  the  second  transformed 
equation  shall  be  of  opposite  sign ;  and  in  the  present  case  this  will  be 
found  to  be  .9. 

721.  If  tKe  coefficient  of  the  first  power  of  the  unknown 
quantity  in  any  transformed  equation  is  zero,  the  next  fig- 
ure of  the  root  may  be  obtained  by  dividing  the  last  term  by 
the  coefficient  of  the  square  of  the  unknown  quantity,  and  tak- 
ing the  square  root  of  the  result. 

For  if  the  transformed  equation  is  y^  +  ay-  +  &  =  0,  it  is 

b 


evident  that,  approximately,  ay-  -\-b  =  0,  or  y  = 

We  proceed  in  a  similar  manner  if  any  number  of  consec- 
utive terms  immediately  preceding  the  last  term  are  zero. 

722.  Horner's  method  may  be  used  to  find  any  root  of  a 
number  approximately ;  for  to  find  the  7ith  root  of  a  is  the 
same  thing  as  to  solve  the  equation  a;"  —  a  =  0. 

723.  If  an  equation  has  two  or  more  roots  which  have 
the  same  integral  part,  the  first  decimal  root-figure  of  each 
must  be  obtained  by  the  method  of  Art.  671  or  673,  or  by 
Sturm's  Theorem, 

If  two  or  more  roots  have  the  same  integral  part,  and  also 
the  same  first  decimal  root-figure,  the  second  decimal  root- 
figure  of  each  must  be  obtained  by  the  method  of  Art.  671 
or  673,  or  by  Sturm's  Theorem ;  and  so  on. 


SOLUTION   OF   HIGHER   EQUATIONS. 


525 


Note  1.  Horner's  method  may  be  used  without  change  to  deter- 
mine successive  figures  in  the  integral,,  as  well  as  in  the  decimal, 
portion  of  the  root. 

Note  2.  If  all  but  one  of  the  roots  of  an  equation  are  known,  the 
remaining  root  may  be  found  by  adding  the  sum  of  the  known  roots  to 
the  coefficient  of  the  second  term,  and  changing  the  sign  of  the  result 
(Art.  639). 

EXAMPLES. 
724.  1.  Find  the  roots  between  1  and  2,  and  —1  and  —2,  of 

2.  Find  the  root  between  5  and  6  of 

a^_(- 20)2 -23a; -70  =  0. 

3.  Find  the  root  between  —  2  and  —  3  of 

—     _-       -4-    -t- 

4.  Find  the  root  between  0  and  1  of 

ar'  +  6iK-  +  10a;-l  =  0. 

5.  Find  the  root  between  —  5  and  —  6  of 

x3-a^-25x  +  81  =  0. 

6.  Find  the  root  between  3  and  4  of 

a;*-10ar^-4a;  +  8  =  0. 

7.  Find  the  root  between  —  2  and  —  3  of 

«*  +  6  a;3  +  12.^2  -  11  .T  -  41  =  0. 

8.  Find  the  root  between  0  and  1  of 

x'  -f  3x^  -3x'  +  19x  -  12  =  0. 
Find  the  real  roots  of  the  following : 

9.  a;2- 2.3a; -1.29  =  0. 
10.    ar'-2>K--a;  +  l  =  0. 

■      11„    a;^_3.^._l  =  0. 

12.  a;3  +  3a;2  +  4a;-f  5  =  0. 
Find  the  approximate  valu( 
17.    i^2.         18     </T7. 


13. 

a;*- 

■12a;  +  7  =  0. 

14. 

a;*- 

,  or^  +  a;  -  2  =  0. 

15. 

a;-'- 

3x2  _  4a; +  13  =  0. 

16. 

x'- 

19x^-23.^-7  =  0. 

3  of  the  following : 

19. 

</5 

K          20.    a/21.5. 

526  COLLEGE   ALGEBRA. 

725.  We  may  now  give  general  directions  for  finding  the 
real  roots  of  any  equation  of  the  form 

with  integral  numerical  coeificieuts  : 

1.  Determine  by  Descartes'  Eule  (Art.  653)  limits  to  the 
number  of  positive  and  negative  roots. 

2.  Find  a  superior  limit  to  the  positive  roots,  and  an 
inferior  limit  to  the  negative  roots  (Arts.  668,  669). 

3.  Divide  the  first  member  by  a;  —  1,  cc  —  2,  a;  +  1,  a;  +  2, 
etc.,  as  explained  in  Art.  689. 

In  this  way  all  the  commensurable  roots,  if  any,  will  be 
found,  and  possibly  all  the  incommensurable  roots  may  be 
located. 

4.  If  the  incoiiimensurable  roots  are  not  all  located,  apply 
Sturm's  Theorem!;  observing  that,  if  the  first  member  and 
its  first  derivative  hav^  a  common  factor,  the  given  equa- 
tion has  multiple  mots  (^.rt.  666). 

5.  Approximate  to  the  decimal  portions  of  the  incom- 
mensurable roots  by  Horner's  Method. 

726.  Newton's  Method  of  Approximation. 

Find  two  numbers,  a  and  b,  one  greater  and  the  other 
less  than  a  root  of  the  equation  (Art.  671),  a  being  alge- 
braically less  than  b. 

Substitute  a-\-y  for  x  in  the  given  equation ;  then  y  is 
small,  and  by  neglecting  the  terms  involving  ?/,  y%  etc.,  an 
approximate  value  of  y  is  obtained  which,  when  added  to  a, 
gives  a',  a  closer  approximation  to  the  value  of  x. 

Substituting  a'+z  for  x  in  the  given  equation,  a  second  ap- 
proximation may  be  obtained  by  the  same  process  as  before. 

Continuing  in  this  way,  the  value  of  the  root  may  be 
obtained  to  any  desired  degree  of  precision. 

Instead  of  substituting  a  -\-y  for  x,  we  may  substitute 
b  —  y,  and  then  proceed  as  above. 


SOLUTION   OF   HIGHER   EQUATIONS.  527 

Exaynple.    Fiud  the  root  between  2  and  3  of  the  equation 

x'-2x-5  =  Q. 
Substituting  y  +  2  for  x,  we  have 

y  +  6f +  102/-1  =  0. 
Whence,  approximately,  lOy  —  1  =  0,  ox  y=  .1. 
Substituting  2  +  2.1  for  x,  we  obtain 

^  +  6.3  2-  + 11.23  z  +  .061  =  0. 

Whence,  approximately,  z  =  —  '         =  —  .005. 

X.±.ajO 

Then,  a;  =  2.1  -  .005  =  2.095,  approximately. 

Note.  Unless  certain  precautions  are  taken,  the  approximation  by 
Newton's  metliod  is  likely  to  fail.  With  reference  to  this  point,  the 
student  may  consult  2'odhunter^s  Theory  of  Equations,  Chap.  XVII. 

For  this  reason,  and  also  because  Horner's  method  is  mucli  shorter, 
Newton's  method  is  of  no  practical  value. 

727.   Approximation  by  Double  Position. 

Find  two  numbers,  a  and  b,  one  greater  and  the  other 
less  than  a  root  of  the  equation /(x)  =  0  (Art.  671),  and  let 
a  be  nearer  to  the  root  than  b. 

If  a  and  b  were  actual  roots,  we  should  have  /(«)  =  0 
and  /(&)  =  0 ;  hence  f(a)  and  f(b)  may  be  considered  as 
the  errors  which  result  from  substituting  a  and  b  in  place 
of  ox 

Although  not  strictly  accurate,  yet,  for  the  purposes  of 
approximation,  we  may  assume  that 

/(a) :  f{b)  =  X  ~  a  :  X  —  b. 

Whence,    /(a) -f{b) : /(a)  =  b-a:x-a  (xVrt.  388). 

Therefore,  '«,_«=  (^:ziM^, 

,   (b  —  a)  f(a)  .^. 


528  COLLEGE   ALGEBRA. 

Example.    Find  the  root  between  4  and  5  of 
f(x)^x'  +  x^  +  x -100  =  0. 

Here  /(4)  =  —  16  and  /(o)  =  55 ;  hence  4  is  nearer  to  the 
root  than  5. 

We  then  have  a  =  4  and  &  =  5. 

Substituting  in  (1),  a;  =  4  H ^^^^  =  4  +  ^  =  4.2  + . 

—  16  —  00  il 

Since  /(4.2)  =  -  4.072  and/(4.3)  =  2.297,  4.3  is  nearer  to 

the  root  than  4.2. 

We  then  have  a  =  4.3  and  b  =  4.2. 

Substituting  in  (1),    x  =  4.3  +  ^^"^^^^^^^^g 

=  4.3-;?|^  =  4.3-.04  =  4.26.  ^ 

Continuing  in  this  way,  the  approximate  value  of  the 
root  may  be  found  to  any  desired  degree  of  accuracy. 

Note.  This  method  of  approximation  lias  the  advantage  of  being 
apphcable  to  any  form  of  equation.  It  may,  tlierefore,  be  applied  to 
the  solution  of  exponential  equations,  and  others  not  in  the  algebraic 
form. 


APPENDIX    I. 

DEMONSTRATION  OF  THE  FUNDAMENTAL  LAWS  OF  ALQEBRA  FOR 
PURE  IMAGINARY  AND  COMPLEX  NUMBERS  (Art.  327). 

Note.  It  will  be  understood  throughout  the  following  discussion 
that  every  .letter  represents  a  positive  real  number  (Art.  318),  unless 
the  contrary  is  expressly  stated. 


728.  Let  XX'  be  a  fixed  straight      ,^j ^_  ^ 

line,  and  0  a  fixed  point  on  the  line.  -i'    ~*^^     0  +«-     ^^ 

We  may  suppose  any  positive  real 
number,  +  a,  to  be  represented  by  a  line  OA,  the  point  J.  being  taken 
a  units  to  the  right  of  0  in  the  line  OX. 

Then  with  the  notation  of  Art.  28,  any  negative  real  number,  —  a, 
may  be  represented  by  a  line  OA',  the  point  A'  being  taken  a  units 
to  the  left  of  0  in  the  line  OA''. 

729.  Since  —a  is  the  same  as  (+ a)  x  (—1),  it  follows  from  Art. 
728  that  the  product  of  +  a  by  —  1  is  represented  by  turning  the  line 
OA,  which  represents  the  number  +  a,  through  two  right  angles,  in  a 
direction  opposite  to  the  motion  of  the  hands  of  a  clock. 

AVe  may  then  regard  —  1,  in  the  product  of  any  real  number  by  —  1, 
as  an  operator  which  turns  the  line  which  represents  the  first  factor 
through  two  right  angles,  in  a  direction  opposite  to  the  motion  of  the 
hands  of  a  clock. 

730.  Consider  the  expression  (+  a)  X «  X  i  (Art.  327,  Note  1). 

By  Art.  7,  this  signifies  that  the  number  +  a  is  multiplied  by  *, 
and  the  result  multiplied  by  i. 

If  we  could  assume  the  Associative  Law  for  Multiplication  (Art.  59) 
to  hold  with  respect  to  the  product  (+a)  XiXh  we  should  have 

(+  a)  X  i  X  i  =  (+  a)  X  i^  =  (+  a)  X  (- 1)  (Art.  328) . 

That  is,  to  multiply  +  a  by  i,  and  then  multiply  the  result  by  i, 
is  the  same  thing  as  to  multiply  +  a  by  —1. 

But  by  Art.  729,  (+  a)  X  (—  1)  is  represented  by  turning  the  line 
which  represents  the  number  +  a  through  two  right  angles,  in  a  direc- 
tion opposite  to  the  motion  of  the  hands  of  a  clock. 


530 


COLLEGE   ALGEBRA. 


We  may  then  define  i,  or  V—  1,  in  the  product  of  any  real  riumbei 
by  i,  as  an  operator  which  turns  the  hue  which  represents  the  real 
number  through  one  right  angle,  in  a  direction 
opposite  to  the  motion  of  the  hands  of  a  clock. 
Hence,  if  XX'  and  YY'  are  fixed  straight 
lines  which  are  perpendicular  to  each  other 
and  intersect  at  0,  and  if  +  a  is  represented 
by  the  line  OA,  where  ^  is  a  units  to  the  right 
of  0  in  the  line  OX,  then  +  ai  may  be  repre- 
sented by  the  line  OB,  where  i>  is  a  units 
above  O  in  the  line  0  Y. 

Again,   with  the   notation  of  Art.  28,  —ai 

may  be  represented  by, the  line  OB',  where  B' 

is  a  units  below  0  in  the  line  01''. 

The  imaginary  numbers  +i  and  —i  are  represented  by  the  lines 

DC  and  00',  where  C  and  C  are,  respectively,  one  unit  above,  and 

one  unit  below  0,  in  the  line  YY'. 

Note.  It  will  be  understood  hereafter  that,  in  any  figure  where 
the  lines  XX'  and  YY'  occur,  they  are  fixed  straight  lines  which  are 
perpendicular  to  each  other  and  intersect  at  0  ;  that  all  positive  or 
negative  real  numbers  are  represented  by  lines  laid  off  to  the  right  or 
left  of  0,  respectively,  in  the  line  A'A'' ;  and  that  all  positive  or  nega- 
tive pure  imaginary  numbers  are  represented  by  lines  laid  off  above 
or  below  0,  respectively,  in  the  line  YY'. 


Y 

■B 

c- 

+ai 

+  i 

^    , 

-i 

c'. 

0-i-a   A 

-ai 
■B' 

y' 

B 


hi 


731.   We  will  now  show  how  to  represent  any  complex  number 
(Art.  327). 

Let  the  number  be   a-\-hi;    and  let  the  real  number  a  be  repre- 
sented by  the  line   OA,  and  the  pure  imaginary  number  hi  by  the 
hue  OB. 

Draw  ^C  equal  and  parallel  to  OB, 
on  the  same  side  of  XX'  as  OB,  and 
join  OC. 
-X  Then  the  complex  number  a  ■\-  hi  is 

represented  by  the  line  OG. 

With  the  notation   of   Art.  28,  the 

complex  number  —  (a  -f  hi)    may  be 

represented   by   the   line    OG ',   where 

OG'  is  equal  in  length  to  OG,  and  is  drawn  in  the  opposite  direction 

from  0. 

In  like  manner,  any  complex  number  whatever  may  be  represented 
by  a  straight  line  drawn  from  0. 


APPENDIX  I.  531 

It  follows  from  Arts.  730  and  731  that  we  may  regard  —1,  in  the 
product  of  any  real,  pure  imaginary,  or  complex  number  by  —1,  as 
an  operator  which  turns  the  line  which  represents  the  first  factor 
through  two  right  angles,  in  a  direction  opposite  to  the  motion  of  the 
hands  of  a  clock.     (Compare  Art.  729.) 

732.  In  the  figure  of  Art.  731,  let  C'A'  be  drawn  peqiendicular  to 
OX' ;  then  the  right  triangles  OA'C  and  OAC  are  equal,  having  the 
hypotenuse  and  an  acute  angle  of  one  equal  to  the  hypotenuse  and  an 
acute  angle  of  the  other,  respectively. 

Then  OA'  and  A'C  are  equal  to  OA  and  AC,  -respectively  ;  that  is, 
OA'  represents  the  real  number  —a,  and  A'C  is  equal  and  parallel  to 
OB',  where  OB'  represents  the  imaginary  number  —hi. 

Therefore  OC  represents  the  complex  number  —  ({ —  bi. 

But  OC  also  represents  —  (a  +  hi)  (Art.  731). 

Whence,  —{a  +  bi)  =  —  a  —  bi. 

733.  The  modulus  of  a  real,  pure  imaginary,  or  complex  number  is 
the  length  of  the  line  which  represents  the  number. 

The  amplitude  is  the  angle  between  the  line  which  represents  the 
number  and  OX,  measured  from  OX  in  a  direction  opposite  to  the 
motion  of  the  hands  of  a  clock. 

If,  for  example,  in  the  figure  of  Art.  731,  the  angle  XOC  is  30°, 
the  amplitude  of  the  complex  number  represented  by  OCis  30^',  and 
the  amplitude  of  the  complex  number  represented  by  OC  is  210°. 

The  modulus  is  always  taken  positive,  and  the  amplitude  may  have 
any  value  between  0°  and  3(30°. 

The  pure  imaginary  numbers  +ai  and  —ai  have  the  modulus  a, 
and  the  amplitudes  90°  and  270°,  respectively  ;  and  the  real  numbers 
+  a  and  —a  have  the  modulus  a,  and  the  amplitudes  0°  and  180°, 
respectively.  

We  have,  in  the  figure  of  Art.  731,  0C=  V  07l'^  +  AC'^=Va^  +  b- ; 
that  is,  the  modulus  of  the  complex  number  a  +  bi  is  Va-  +  b'^  I  and 
this  is  also  the  modulus  of  each  of  the  complex  numbers  ±a±bi. 

Whatever  number  is  represented  by  a,  the  amplitude  of  —a  is 
always  equal  to  the  amplitude  of  a  increased  by  180°. 

Note.    We  may  regard  zero  as  having  the  modulus  zero. 

Addition  and  Subtraction  of  Imaginary  Numbers. 

734.  The  representation  of  a  complex  number,  as  explained  in  Art, 
731,  shows  that  the  result  of  adding  a  pure  imaginary  to  a  real  num- 
ber may  be  represented  by  a  straight  line  drawn  from  0. 


532 


COLLEGE   ALGEBRA. 


We  will  now  show  how  to  represent  the  result  of  adding  h  to  a, 
where  a  and  h  represent  any  two  real,  pure  imaginary,  or  complex 
numbers. 

Let  a  be  represented  by  OA,  and  h  by  OB. 

Draw  AC  equal  and  parallel  to  OB.,  in  such  a 
way  that  C  shall  be  in  the  same  direction  from 
A  that  B  is  from  0. 

Then  the  result  of  adding  6  to  «  is  represented 
by  the  line  OC. 

That  is  (Art.  -5),  a  +  hi?,  represented  by  OC. 

Note  1.  The  above  construction  holds  equally  when  OA  and  OB 
lie  in  the  same  direction,  or  in  opposite  directions,  from  0. 

Note  2.  The  form  of  addition  exemplified  in  the  above  construc- 
tion is  known  as  Geometric  Addition. 

In  like  manner,  the  result  of  adding  any  number  of  real,  pure 
imaginary,  or  complex  numbers  may  be  represented  by  a  straight  line 
drawn  from  0. 

735.   In  the  figure  of  Art.  734,  draw  BC. 

By  Geometry,  OACB  is  a  parallelogram,  and  therefore  BC  is  equal 
and  parallel  to  OA. 

Then  OC  represents  the  result  of  adding  a  to  h. 

But  OC  also  represents  the  result  of  adding  h  to  a. 

Whence,  a-\-b=b  +  a.     (Compare  Art.  36.) 

The  above  result  holds  if  either  of  the  letters  a  and  h  repre- 
sents the  sum  of  any  number  of  real,  pure  imaginary,  or  complex 
numbers. 


736. .  We  shall  define  the  subtraction  of  b  from  a,  where  a  and  b 
represent  any  two  real,  pure  imaginary,  or 
complex  numbers,  as  the  process  of  finding  a 
number  such  that,  when  h  is  added  to  it,  the 
sum  shall  be  equal  to  a.     (Compare  Art.  41.) 

Let  a  be  represented  by  OA ,  and  b  by  OB ; 
and  complete  the  parallelogram  OB  AC. 

By  Art.  734,  OA  represents  the  result  of 
adding  the  number  represented  by  OB  to  the 
number  represented  by  OC;  that  is,  if  b  is 
added  to  the  number  represented  by  OC,  the 
sum  is  equal  to  a. 
b  is  represented  by  the  line  OC. 


APPEXDIX  I.  533 

737.  In  the  figure  of  Art.  73G,  let  OB  be  produced  to  B',  making 
OB'  equal  to  OB. 

Then  since  J.O  is  equal  and  parallel  to  OB'^  00  represents  the 
result  of  adding  the  number  represented  by  OB'  to  the  number  repre- 
sented by  OA. 

J3ut  by  Art.  731,  the  number  —6  is  represented  by  OB'. 

Whence,        a  —  6  =  a  +  (—  6).     (Compare  Art.  42.) 

Again,  the  result  of  adding  —  6  to  fe  is  represented  by  a  line  joining 
O  to  the  extremity  of  a  line  drawn  from  B  towards  0,  equal  and 
parallel  to  OB' . 

That  is, ,  &  +  (-  h)  =  0.     (Compare  Art.  39.) 

738.  Let  a,  6,  and  c  be  any  three  real,  pure  imaginary,  or  complex 
numbers,  represented  by  the  lines  0^4,  OB., 

and  OC,  respectively.  ^ ^ 

Complete  the  parallelograms  OADB  and  /^^-^^      ,^T^\ 

OB  EC ;  then  OD  represents  a  +  b  (Art.  734)  /         /^"t-'-Os 

and  in  like  manner  OE  represents  b  +  c.  /       /         ''^--^  ^ 

Complete  the  parallelogram  OBFC ;  then         /  ^'^^.^"-'''''''^^^ ' 

a  +  b  +  c,  being  the  result  obtained  by  add-      O^C^i^ Hc'^       i 

ing  c  to  ffl+6  (Art.  5),  is  represented  by         \    "~^s<-^   \     / 
the  Ime  OF.  .  V^^-lljl-V' 

Join  A  F  and  EF.  ^  D 

Since,  by  construction,  DF  and  BE  are 
equal  and  parallel  to  OC,  they  are  equal  and  parallel  to  each  other, 
and  BDFE  is  a  parallelogram. 

Therefore  EF  is  equal  and  parallel  to  BD,  and  consequently  to  OA ; 
and  hence  OAFE  is  a  parallelogram. 

Then  OF  represents  the  result  of  adding  the  number  represented 
by  OE  to  the  number  represented  by  OA  ;  that  is,  OF  represents 
a+(&  +  c). 

But  OF  also  represents  a  +  ft  +  c. 

Hence,  a  +  (5  +  c)  =  a  +  6  +  c.     (Compare  Art.  37. ) 

The  above  result  holds  if  any  or  all  of  the  letters  a,  &,  and  c  repre- 
sent the  sum  of  any  number  of  real,  pure  imaginary,  or  complex 
numbers  ;  and  hence  the  Associative  Law  for  Addition  (Art.  37)  holds 
when  any  or  all  of  the  numbers  involved  are  pure  imaginary  or 
complex. 

739.  "We  will  now  prove  that  the  Commutative  Law  for  Addition 
(Art.  36)  holds  when  any  or  all  of  the  numbers  involved  are  pure 
imaginary  or  complex. 


534  COLLEGE   ALGEBRA. 

Consider  the  expression 

a  +  b  +  -  +  c  +  d  +  e+f+-  +  g, 
where  a,  h,  c,  etc.,  are  any  real,  pure  imaginary,  or  complex  numbers. 
By  Art.  738, 

a  +  b  +  ---  +  c+  cl+e  +f+  —  +  g 
=  a+b  +  -  +  c+id  +  e)+f-\--  +  g 
=ra+b  +  ---  +  c+(e  +  d)+f+--  +  g  (Art.  735) 
=  a  +  6  +  -  +  c+   e  +  d  +f+-  +  g  (Art.  738). 

Tliat  is,  any  two  consecutive  terms  of  an  expression  may  be  inter- 
changed without  altering  the  value  of  the  expression. 

Now  by  successive  interchanges  of  consecutive  terms,  the  terms  of 
an  expression  may  be  written  in  any  order  whatever. 

Hence  the  Commutative  Law  for  Addition  holds  when  any  or  all  of 
the  numbers  involved  are  pure  imaginary  or  complex. 

Note.  It  follows  from  what  has  already  been  proved,  that  the 
results  in  Arts.  43  to  49  inclusive  hold  for  any  pure  imaginary  or  com- 
plex values  of  the  letters  involved. 

Multiplication  of  Imaginary  Numbers. 

740.   Since  -f  ai  may  be  written  (-f  1)  x  (+  aO,  the  product  of  -f  1 
by  -I-  ai  is  represented  by  turning  the  line  OA, 
which  represents  the  number  -f  1,  through  one 
S  "  right  angle,  in  a  direction  opposite  to  the  motion 

+«-*  of  the  hands  of  a  clock,  and  multiplying  the 


X- 


^^  result  by  a 

— a-. > 


■X  And  since  —  ai  may  be  -^ATitten  (+ 1)  X(—  ai), 

the  product  of  -F  1  by  —  ai  is  represented  by  a 

-ai  line  equal  in  length  to  that  which  represents 

.j3'  the  product  of  -1- 1  by  -t-ai,  but  drawn  in  tho 

opposite  direction  from  0. 
•*•  This  suggests  the  following  : 

The  product  of  any  real,  pure  imaginary,  or  complex  number  by 
-\-ai  may  be  represented  by  turning  the  line  which  represents  the 
number  through  one  right  angle,  in  a  direction  opposite  to  the  motion 
of  the  hands  of  a  clock,  and  multiplying  the  result  by  a. 

The  product  of.  any  real,  pure  imaginary,  or  complex  number  by 
—  ai  may  be  represented  by  aline  equal  in  length  to  the  line  which 
represents  its  product  by  -^ai,  but  drawn  in  the  opposite  direction 
from  0. 


APPENDIX   I. 


535 


^ 

, 

0 

y\  , 

r^'y 

/^ 

+1.1 

741.    Since  a  +  6i  may  be  written  (+  1)  X  (a  +  hi),  the  product  oi 
+  1  by  a-\rhi  is  represented  by  turning  the 
line  OA,  which  represents  the  number  + 1, 
through  an  angle  equal  to  the  amplitude  of 
rt+6t  (Art.  733),  in  a  direction  opposite  to 

the  motion  of  the  hands  of  a  clock,  and  mul-         ' --m/    i  ■        ^l, 

tiplying  the  result  by  the  modulus  of  a  +  hi. 

And   since    —  (a  +  hi)    may   be    written 
(+  1)  X  (—a  —  60  (Art.  732),  the  product  of 
+  1  by  —  (a  +  fei)  is  represented  by  a  line 
equal  in  length  to  that  which  represents  the  product  of  +1  by  a-\-hi, 
but  drawn  in  the  opposite  direction  from  0. 

This  suggests  the  following : 

If  a  and  h  are  any  positive  or  negative  real  numbers,  the  product 
of  any  real,  pure  imaginary,  or  complex  number  by  a-^hi  may  be 
represented  by  turning  the  line  which  represents  the  number  through 
an  angle  equal  to  the  amplitude  of  a+6i,  in  a  direction  opposite  to 
the  motion  of  the  hands  of  a  clock,  and  multiplying  the  result  by  the 
modulus  of  a  +  hi. 

The  product  of  any  real,  pure  imaginaiy,  or  complex  number  by 
—  {a-\-hi)  may  be  represented  by  a  line  equal  in  length  to  the  line 
which  represents  its  product  by  a  +  hi,  but  drawn  in  the  opposite 
direction  from  0. 


742.    Let  a  and  h  be  any  two  real,  pure  imaginary,  or  complex 
numbers,  represented  by  the  lines  OA  and 
Oi?,  respectively. 

Then  the  result  of  multiplying  a  by  &  is 
represented  by  OC,  where  the  angle  XOC  is 
the  sum  of  the  angles  XOA  and  XOB,  and 
OC  is  equal  to  OA  x  OB. 

That  is  (Art.  7),  ah  is  represented  by  OC. 

In  like  manner,  the  product  of  any  number 
of  real,  pure  imaginary,  or  complex  numbers 
may  be  represented  by  a  straight  line  drawn 
from  0. 

It  is  evident  from  the  above  that  the  modulus  of  the  product  of  two 
or  more  numbers  is  the  product  of  their  moduli,  and  that  its  amplitude 
is  the  sum  of  their  amplitudes. 

Note.  The  form  of  multiplication  exemplified  in  the  above  con- 
.struction  is  known  as  Geometric  Multiplication. 


636  COLLEGE   ALGEBRA. 

743.  With  the  figure  and  notation  of  Art.  742,  bxa  is  represented 
by  turning  OB  through  an  angle  equal  to  XOA,  in  a  direction  opposite 
to  the  motion  of  the  hand.s  of  a  clock,  and  multiplying  the  result  by  OA. 

That  is,  6  X  a  is  represented  by  OC. 

It  is  evident  from  this  that  axb^bxa.     (Compare  Art.  58.) 
The  result  holds  if  either  or  both  of  the  letters  a  and  b  represent  the 
product  of  any  number  of  real,  pure  imaginary,  or  complex  numbers. 

744.  In  the  figure  of  Art.  742,  let  OC  be  produced  to  C",  making 
OC"  equal  to  OC. 

Then  by  Arts.  740  and  741,  a  X  (— fc)  is  represented  by  OC. 
But  OC  also  represents  —ah. 

Therefore,  aX(-b)  =  -ab.  (1) 

Again,  let  OA  be  produced  to  A',  making  OA'  equal  to  OA. 

Then  —  a  is  represented  by  OA' ;  and  consequently  (—  a)  X  &  is 
represented  by  turning  OA'  through  an  angle  equal  to  XOB,  in  a 
direction  opposite  to  the  motion  of  the  hands  of  a  clock,  and  multiply- 
ing the  result  by  the  modulus  of  b ;  that  is,  (—a)xb  is  represented 
by  OC. 

Therefore,  (—  a)  X  &  =  —  ab.  (2) 

By  (2),  (_a)x(-6)  =  -aX(-6) 

=  _(_a6),by  (1) 
=  +  ab. 

Hence  the  results  in  Art.  56  hold  if  either  or  both  of  the  letters 
involved  represent  pure  imaginary  or  complex  numbers. 

745.  Let  «,  b,  and  c  be  any  three  real,  pure  imaginary,  or  complex 
numbers. 

Then  ax(,bxc)  is  represented  by  turning  the  line  which  repre- 
sents a  through  an  angle  equal  to  the  sum  of  the  amplitudes  of  b  and  c 
(Art.  742),  in  a  direction  opposite  to  the  motion  of  the  hands  of  a 
clock,  and  multiplying  the  result  by  the  product  of  the  moduli  of  b 
and  c. 

Again,  axbxc,  being  the  result  of  multiplying  axb  by  c  (Art.  7), 
is  represented  by  turning  the  line  which  represents  a  through  an  angle 
equal  to  the  amplitude  of  b,  in  a  direction  opposite  to  the  motion  of 
the  hands  of  a  clock,  and  multiplying  the  result  by  the  modulus  of  b  ; 
and  then  turning  the  resulting  line  in  the  same  direction  through  an 
angle  equal  to  the  amplitude  of  c,  and  multiplying  the  result  by  the 
modulus  of  c. 


APPENDIX   I. 


537 


It  is  evident  from  this  that 

axibxc)  =  axbxc.     (Compare  Art.  59.) 

The  above  result  holds  if  either  of  the  letters  a,  b,  and  c  represents 
the  product  of  any  number  of  real,  pure  imaginary,  or  complex  num- 
bers ;  and  hence  the  Associative  Law  for  Multiplication  (Art.  59) 
holds  when  any  or  all  of  the  numbers  involved  are  pure  imagmary  oi 
complex. 

746.   We  will  now  prove  that  the  Commutative  Law  for  Multipli- 
cation (Art.  58)  holds  when  any  or  all  of  the  numbers  involved  are 
pure  imaginary  or  complex. 
Considei  the  expression 

axbx---  XcXdXeXfX---  Xg, 
where  «,  b,  c,  etc.,  are  any  real,  pure  imaginary,  or  complex  numbers. 
By  Art.  745, 

axbx---  XcX    dxe    Xf<---Xg 
=  axbx---  XcX(dXe)  xfX  ■■■  X  (J 
=  axbx--  Xcx(ex  d)  X/X  •••  X  g    (Art.  74-3) 
=  axbx  — XCX    exd    XfX---Xg    (Art.  745). 
That  is,  any  two  consecutive  factors  of  a  product  may  be  inter~ 
changed  without  altering  the  value  of  the  expression. 

Now  by  successive  interchanges  of  consecutive  factors,  tlie  factor.s 
of  a  product  may  be  written  in  any  order  whatever. 

Hence  the  Commutative  Law  for  Multiplication  holds  when  any  or 
all  of  the  numbers  involved  are  pure  imaginary  or  complex. 


747.  Let  a,  6,  and  c  be  any  three     ^^ 
real,  pure  imaginary,  or  complex  num- 
bers, represented  by  the  lines  OA,  OB, 
and  OC,  respectively. 

Complete  the  parallelogram  OBDC; 
then  by  Art.  734,  6  -f  c  is  represented 
by  OD. 

Hence,  by  Art.  742,  aX{b-\-  c)  is 
represented  by  OE,  where  the  angle 
XOE  is  the  sum  of  the  angles  XOA 
and XOD,  and  OE  is  equal  to  OAxOD. 

Again,  ab  is  reiiresented  by  OF,wliere 
the  angle  XOF  is  the  sum  of  the  angles  XOA  and  XOB,  and  0  P  is  equal 
to  OA  X  OB  ;  and  ac  is  represented  by  OG,  where  the  angle  XOG  is 
the  sum,  of  the  angles  XOA  and  .YOC,  and  OCr  is  equal  to  OAxOC. 


iX'Axr 

Y 

Y 

\     CI   \\  1 

h 

^-^•1 

X'              0 

y' 

A' 

538  COLLEGE   ALGEBRA. 

Join  EF,  EG,  BF,  and  CG. 

.      ,.  OE      OF     OG      r>A 

By  construction,         ^  =  ^  =  ^-^  =  0A 

Whence,  by  alternation  (Art.  385), 

9^  =  ^  and^  =  — 
OF      OB  OG     OC' 

Again,  the  angles  EOD,  BOF,  and  COG  are  equal,  since,  by  con- 
struction, each  is  equal  to  the  angle  XOA. 

Therefore  the  angles  EOF  and  BOD  are  equal  ;  for  EOF  is  the  sum 
of  BOF  and  EOD,  and  BOD  is  the  sum  of  DOF  and  BOF,  which  is 
equal  to  EOD. 

In  hke  mamier,  the  angles  EOG  and  COD  are  equal. 

Therefore  the  triangles  EOF  and  BOD  are  similar,  as  also  are  the 
triangles  EOG  and  COD  ;  for,  by  Geometry,  two  triangles  are  similar 
when  they  have  an  angle  of  one  equal  to  an  angle  of  the  other,  and  the 
including  sides  proportional. 

Then  the  figure  OFEG  is  similar  to  OB  DC,  and  hence  OFEG  is  a 
parallelogram. 

Therefore  OE  represents  the  sum  of  the  numbers  represented  by 
OF  and  OG  ;  that  is,  OE  represents  ab  +  ac. 

But  OE  also  represents  «  x  (6  +  c). 

Therefore,  a  x  (b  +  c)  =^  ab  +  ac. 

Hence  the  Distributive  Law  for  Multiplication  (Art.  60)  holds  when 
any  or  all  of  the  numbers  involved  are  pure  imaginary  or  complex. 

Division  of  Imar/inary  Numbers. 

748.  We  shall  define  the  quotient  of  a  divided  by  h,  where  a  and  h 
are  any  two  real,  pure  imaginary,  or  complex  numbers,  as  the  process 
of  finding  a  number  such  that,  when  it  is  multiplied  by  b,  the  product 
shall  be  equal  to  a.     (Compare  Art.  67.) 

y  Let  o  be  represented  by  OA,  and  b 

by  OB. 

Draw  OC  so  .that  the  angle  XOC  shall 
li  be  equal  to  the  angle  XOA  minus  the 

-X  angle  XOB,  and  OC  equal  to  — • 

^^  Then  the  angle  XOA  is  equal  to  the 

angle  XOC  plus  the  angle  XOB,  and  OA  is  equal  to  OC  x  OB  ;  and 
hciice,  by  Art.  742,  OA  represents  the  product  of  the  number  repre- 
Rcntt'd  by  Or,';uid  the  number  rrpri'seiitod  by  07?. 


APPENDIX   I.  539 

That  is,  if  the  number  represented  by  OC  is  multiplied  by  6,  the 
product  is  equal  to  a. 

Therefore,  -  is  represented  by  0  C. 
b 

It  is  evident  from  the  above  that  the  modulus  of  the  quotient  of  two 
numbers  is  equal  to  the  modulus  of  the  dividend  divided  by  the  modu- 
lus of  the  divisor,  and  its  amplitude  equal  to  the  amplitude  of  the 
dividend  minus  the  amplitude  of  the  divisor. 

Note.  It  follows  from  what  has  already  been  proved,  that  the 
results  in  Arts.  68  to  72  inclusive  hold  for  any  pure  imaginary  or  com- 
plex values  of  the  letters  involved. 

749.  We  define  y/a,  where  a  is  any  real,  pure  imaginary,  or  com- 
plex number,  as  a  number  whose  ?tth  power  is  equal  to  a  (Art.  121). 

Let  a  be  represented  by  OA. 

Draw  OB  so  that  the  angle  XOB  shall  be  equal  to  one-nth  of  the 
angle  XOA,  and  the  length  of  OH  equal  to  the 
nth  root  of  the  modulus  of  a. 

Then  the  angle  XOA  is  ?i  times  the  angle 
XOB,  and  the  modulus  of  a  is  the  nth  power  of 
the  length  of  OB  ;  and  hence,  by  Art.  742,  OA 
represents  the  nth  power  of  the  number  repre- 
sented by  OB. 

Therefore,  \^a  is  represented  by  OB. 

750.  We  have  proved  that  every  result  in  Chapter  II.  holds  when 
any  or  all  of  the  numbers  involved  are  pure  imaginary  or  complex  ; 
and  therefore  every  statement  or  rule,  in  Chapters  III.  to  XVI.  inclu- 
sive, in  regard  to  expressions  where  any  letter  involved  represents  any 
real  number  whatever,  holds  equally  when  this  letter  represents  any 
pure  imaginary  or  complex  number.     (Compare  Art.  321.) 

751.  It  is  evident  from  Arts.  734,  736,  742,  748,  and  749,  that  any 
expression  which  is  the  result  of  any  finite  number  of  the  following 
operations  performed  upon  one  or  more  real,  pure  imaginary,  or  com- 
plex numbers,  may  be  represented  by  a  straight  line  : 

1.  Addition  or  Subtraction.  2.  Multiplication  or  Division.  3.  Rais- 
ing to  any  power  whose  exponent  is  a  rational  number  (Art.  209). 
4.  Extracting  any  root. 

That  is,  any  such  number  can  be  expressed  in  the  form  a+  hi, 
where  a  and  h  are  real  numbers,  either  or  both  of  which  may  be  zero. 


640  COLLEGE   ALGEBRA. 

APPENDIX  11. 

CAUOHT'S  PROOF  THAT  EVERY  EQUATION  HAS  k   ROOT. 

752.    To  prove  that  Va  +  bi,  where  a  and  b  are  real  numbers,  can 
be  expressed  in  the  form  c  +  di,  where  c  and  d  are  real  numbers. 

Squaring  the  equation  Va  +  bi  =  c  +  di,  we  have 

a  +  bi  =  c^-\-  2  cdi  —  d^. 

Whence  (Art.  338),  c"-  -  cV  =  a,  (1) 

and  2  cd  =  b.  (2) 

Squaring  (1 ) ,  c*  -  2  c'^  cZ^  +  d*  =  a^  (3) 

Squaring  (2),  4c'^d^=b'^.  (4) 

Adding  (3)  and  (4),     c*  +  2  c^d^  +  d^  =  a^  +  h\ 


Whence,  c'^  +  d^=  Va^  +  b'^.  (5) 

The  upper  sign  must  be  taken  before  the  radical  in  equation  (5)  ; 
for  since  by  hypothesis  c  and  d  are  real  number^,  c-  +  d-  is  positive. 


Adding  (1)  and  (5),                        2  c-  =  «  +  Va-  -r  0-.  (0) 

Subtracting  (1)  from  (5),               2  cZ-  =  —  a  +  Va-  +  h'.  (7) 

Equations  (6)  and  (7)  show  that  c-  and  d-  are  positive,  and  there- 
fore c  and  d  are  real. 

753.   To  prove  that,  if  n  is  a  positive  integer,  each  of  the  equations 
X"  =  ±  1 ,  and  X"  -  rb  i 
has  a  root  of  the  form  a  +  hi,  where  a  and  &  are  real  numbers,  either 
of  which  may  be  zero. 

Case  I.     a;»  =  l. 

It  is  evident  that  1  is  a  root  of  this  equation. 

Case  II.     x"  =  —  1,  where  n  is  odd. 

It  is  evident  that  —  1  is  a  root  of  this  equation. 

Case  III.     %"■=  —\,  where  n  is  even. 

Let  ra  =  2  m,  where  m  is  a  positive  integer ;  then,  x^"  =  —  1. 
Extracting  the  square  root  of  each  member,  x'"  =  ±  i. 
The  latter  forms  are  included  in  the  four  following  cases. 


APPENDIX   II.  541 

Case  IV.     a;"  =  i,  where  n  is  odd. 

If  m  is  a  positive  integer,  i*"^-*-^  =  i  (Art.  333)  ;  hence,  if  n  is  of  the 
form  4  j/«  +  1,  Jis  a  root  of  the  equation. 

Ao-ain,  (— i)4'n+3=  _  j^m+s^  _  (_j)  (Art.  333)  =  i ;  hence,  if  n  is 
of  tlie  form  4  hi  +  3,  -  i  is  a  root  of  the  equation. 

Case  V.     x"-  =  i,  wliere  n  is  even. 

Let  n  -=  2'>p,  where  p  is  an  odd  integer ;  then,  x'^^p  =  i. 

Let  x^  =  y;  tlien  2/P  =  i,  and,  by  Case  IV.,  y  =  i  or  —i  according 
as  p  is  of  the  form  4  ni  +  1  or  4  jw  +  3 ;  that  is,  a;-'  =  i  or  -  i. 

The  value  of  x  may  be  obtained  from  this  equation  by  q  successive 
extractions  of  the  square  root ;  and  since  it  has  been  proved  that  the 
square  root  of  a  +  hi,  where  a  and  b  are  real,  can  be  expressed  in  the 
same  form,  it  follows  that  x  can  be  expressed  in  the  form  a  +  hi. 

Case  VI.     x"-=  —  i,  where  n  is  odd. 

By  Art.  333,  (— j)''"'+i  =  —  i-*"'+i=  —  i ;  hence,  if  n  is  of  the  form 
4  j?t  +  1,  —i  is  a  root  of  the  equation. 

Again,  ii"'+^=  —  i ;  hence,  if  n  is  of  the  form  4?n  +  3,  i  is  a  root. 

Case  VII.     a;"=  —  t,  where  n  is  even. 

As  in  Case  V.,  x  may  be  obtained  in  the  form  a  +  hi. 

754.    We  will  now  consider  the  general  case. 

To  prove  that  the  general  equation  of  the  ?«th  degree 

x»  +  i)iX"-i+i>,a;«-^+  •••  +p«-ix  +  p„=0  (1) 

has  a  root  of  the  form  a  +  hi,  where  a  and  h  are  real  numbers. 
Substituting  a  +  hi  for  x  in  (1)  we  have 

(a  +  &0"  +  Pi  («  +  ^0"'^  +  •••  +  Pn~\  («  +  60  +  Pn  =  0. 
Expanding  by  the  Binomial  Theorem,  and  collecting  together  the 
real  and  imaginary  terms,  we  shall  have  a  result  of  the  form 

C/+Fi  =  0,  (2) 

where  U  and  V  are  real  numbers. 

Transposing  Vi  in  (2),  and  squaring  both  members,  we  have 

m  =  -  F^  or  U^  +  V^  =  0. 
We  have  then  to  prove  that  such  real  values  may  be  found  for  a 
and  h  as  will  make  U-  +  F^  =  0. 

As  a  and  &  change  in  value,  U  and  F  also  change  ;  and  if  U-  +  F- 
cannot  become  zero  for  any  values  of  a  and  h,  there  must  be  some  pos- 
itive real  number  which  is  the  least  value  that  U'^+V'^  can  assume. 


542  COLLEGE    A1.GEBKA. 

Let  a  and  /S  be  the  values  of  a  and  b,  respectively,  for  wlilch  U^  +  V 
has  this  minimum  value. 

Let  P+  Qi  be  the  value  of  the  first  member  of  (1)  when  a+  pi  is 
substituted  for  x  ;  then  P^  +  Q^  is  the  minimum  value  of  U'^  +  V'^. 

Writing  o  +  yOi  +  h  in  place  of  x  in  (1),  we  have 

(a  +  /Si  +  hy+l\  (a  +  Hi  +  /0«^1  +  •••  +P«-1  (a  +  Hi  +  /i)  +  i5„  =  0. 

Exisanding  by  the  Binomial  Theorem,  and  arranging  the  result  in 
ascending  powers  of  h, 

(a+/30"+P^(a+;30«-l+  ...  +  ;,„  _i  (a  +  ^i) +^„ 

+  h  In  (a  +  Hiy-^  +  Pi  («  -  1)  (a  +  /aO''-^  4.  ...  j^  p^^j-j 
+  (terms  involving  7t',  /i^,  ...,  A")  — .  0.  (3) 

The  first  line  of  (3)  is  equal  to  P  +  Qi. 

The  coefficients  of  some  of  the  powers  of  h  may  be  zero  ;  but  they 
cannot  all  be  zero,  since  the  coefficient  of  ft"  is  unity. 

Let  /t™  be  the  lowest  power  of  h  whose  coefficient  is  not  zero  ;  and 
denote  its  coefficient  by  /?  +  -S'j,  where  R  and  S  are  not  both  zero. 
Then  (3)  becomes        P  +  (2»  +  (P  +  Si)  h'^ 

4-  (terms  involving  powers  of  h  higher  than  the  mih.)  =  0. 
Let  this  be  denoted  by        P'  +Q'i=^Q.  (4) 

Now  let  h  =  ct,  where  c  is  a  positive  real  number,  and  t  a  root  of 
the  equation  i™  =  1  or  ("^  =  —  1 . 

By  Art.  753,  t  is  in  cither  case  a  number  of  the  form  a  +  hi. 

Then,  pi^  qti  =  p^  Qi  ±  (p  +  Si)  C"  +  -  . 

Whence  (Art.  .338),  P'  ^  P±  lie"'  +  ••• , 

and  Q'=r  Q±Sc-"  +■■•■ 

Therefore,  P'2  +  Q'^=  p-^+ Q^±2  {PR  +  QS)  c"  +  •••  • 

That  is,      P'2  +  Q'2  -  pi  -  Q2  =  ±  2  (PP  +  Q^S)  C" 

+  (terms  involving  powers  of  c  higher  than  the  ?7ith).  (5) 

If  PR  +  QS  is  not  zero,  c  may  be  taken  so  small  that  the  sign  of 
the  second  member  will  be  the  same  as  that  of  ±  2  {PR  +  Q*S')c"». 

Hence,  if  PR+QS  is  positive,  the  sign  of  P'- +  Q'^  —  P'^  —  Q'^ 
may  be  made  negative  by  taking  i"*  =  —  1  ;  and  if  PR  +  QS  is  nega- 
tive, the. sign  of  P'^  +  Q'^  —  P^  —  Q-  may  be  made  negative  by  taking 
«-  =  +  L 

'■J'hus,  in  either  case,  P'-  +  Q'-  can,  by  properly  choosing  c  and  t,  be 
madi;  less  than  P-  +  Q-. 


APPENDIX   II.  54a 

If  PR  +  QS=  0,  let  «>"  =  ±  i  in  (4). 

By  Art.  753,  t  is  in  either  case  a  number  of  tlie  form-  a  +  hi. 
Then,  P'+q'i  =  P+Qi±{R  +  Si)ic^  +  - 

=  P  +  Qi±{Bi-S)c"'+-' 
Whence,  F'  =  PT  Sc"  +  ■■■, 

and  ■  Q'=Q±Rc"^+-- 

Therefore,  P''  +  Q''  =  P' +  Q^±2  (QB-  PS)  c-  +  •.• » 

That  is,     P''  +Q'^-P-'-Q-'  =  ±  2  {QR-PS)  c- +  -  ■ 

Now,     (PA'  +  Qsr  +  (QR-PSy'  =  P'R'  +  Q'S-'  +  q^  R^  +  P'^S-' 

And  since,  by  hypotliesis,  P-^  +  Q-  is  not  zero,  and  R  and  S  are  not 
both  0,  it  follows  that  {PR+  QSy  +(QR- PSy  is  not  zero. 

But  PR  +  QS=  0,  and  hence  QR  -  PS  is  not  zero. 

Therefore,  if  c  is  taken  sufficiently  small,  the  sign  of  P'-+  Q'-—P-—Q^ 
will  be  the  same  as  the  sign  of  ±2{QR-PS)c'"  ;  and  we  can  ensure 
that  this  sign  shall  be  negative  by  taking  r  =  -  i  when  QR  -PS  is 
positive,  and  t"'=i  when  QR  —  PS  is  negative. 

Thus,  by  properly  choosing  h,  P'^  +  Q'^  may  be  made  less  than 
P-  +  Q^ ;  that  is,  a  value  of  U'  +  F-  may  be  obtained  which  is  less 
than  P2  +  Q^,  and  the  latter  is  not  a  minimum  value  of  U^  +  V\ 

Hence,  no  positive  real  number  can  be  a  minimum  value  of 
U-  +  V'^ ;  and  therefore  values  of  a  and  b  can  be  found  which  will 
make  f/^  +  F'^  =  0. 

We  will  now  prove  that  the  values  of  a  and  b  which  make 
U-  +  F-  =  0  are  finite. 

The  first  member  of  (1)  may  be  written 

\  X         X^  X"J 

Putting  a  +  bi  in  place  of  x,  we  have 

^7+  vi=(a  +  bi)"[i  +  -^  +  ^fW+  •••  +  7— rhrnT  (^^ 

^  L       a+bi      (a  +  bi)-  {a  +  bi)"J 

Consider  the  term 


2),-       _        pr(a  —  biy 


ia  +  bi)'-      [(a  +  bi){a-bi)y      (a^+fe-) 


(rt  —  50'' 


_P^^{ar-ra-^bi-'^^^;^a^-'b^~+  - 
a-^  +  b-^yl 


(a-^  +  b-^yl  \2 

=  Ar  +  P,i,   say. 


544  COLLEGE   ALGEBRA. 

Now,      A=^^^wk-'%^«^^^^^+-l 

■'■[(#-'"S"(.T!)' ■(?:)■•■■■]■ 

It  is  evident  from  tliis  that,  when  a  and  b  are  indefinitely  increased, 
Ar  is  indefinitely  diminished ;  and  in  like  manner  it  may  be  shown 
that  B,  is  indefinitely  diminished  when  a  and  b  are  indefinitely  in- 
creased. 

Thus  (6)  may  be  written 

U+  Vi  =(a  +  ?)i)"  [1  +  ^1'  +  S'i],  (7) 

where  A'  and  B'  are  indefinitely  diminished  when  a  and  b  are  indefi- 
nitely increased. 

If  a  —  bi  is  substituted  for  x  in  (1),  we  shall  have  a  result  which 
may  be  obtained  from  (7)  by  simply  changing  the  signs  of  the  terms 
involving  i ;  thus, 

U-Vi=(a-biy[l  +  A>-B'i].  (8) 

Multiplying  (7)  and  (8),  m  +  V^  =  (a^  +  b'^)"  [(1  +  A'^  +  B''^].  (9) 

The  second  member  of  (9)  increases  indefinitely  when  a  and  b  are 
indefinitely  increased;  for  the  factor  (a- +  fe"^)"  increases  indefinitely, 
and  the  factor  (1  +  A')'^  +  B'^   approaches  the  limit  1. 

Hence,  U"'^  +  V^  cannot  be  zero  when  a  and  b,  or  either  of  them, 
are  indefinitely  increased ;  and  therefore  the  values  of  a  and  b  which 
make  U'^  +  V^  =  0  are  finite. 

755.  The  demonstration  of  Art.  754  holds  equally  whether  the 
coefhcients  of  the  terms  in  equation  (1)  are  real  or  imaginary. 

It  follows  from  the  above  that  y/—  a,  where  n  is  any  even  integer 
and  a  a  positive  real  number,  and  v'a  +  bi,  where  n  is  any  positive 
integer  and  a  and  6  any  real  numbers,  can  be  expressed  in  the  form 
c  +  di,  where  c  and  d  are  real  numbers. 

That  is,  any  even  root  of  a  negative  real  number,  or  any  root  of 
a  pure  imaginary  or  complex  number,  can  be  expressed  as  a  pure 
imaginary  or  complex  number. 

(^Compare  Art.  33G  ;  also,  Appendix  I.,  Art.  751.) 


ANSWEES. 


Note.    In  the  following  collection  of  answers,  all  those  are  omitted 
I'hich,  if  given,  would  destroy  the  utility  of  the  example. 


Art.  84  ;  page  28. 

1.    ^x-^+Sx+2.         2.    Sa  +  Sb+Sc  +  Sd.  3.   5fl3 -3n62 -46^. 

4.    a^-ix^.  5.    -3x3-5a:2^  +  8.r^2  +  2/. 

6.  5x^+Gxy-l  y-^-Gx-~  y+6.     7.    3.r5-5;ri  — .r^- ll.r2+ 4.r-2. 

8.  4x3  +  9.r2(/-4.ry2_3y3.      9.    4«4  _i_  7^3  _  2a  _  4.      10.    2x  +  4i/. 
11.    a-c.       12.    -3a-l.       13.   4,r-2.      14.   Gm  +  2.      15.    a +  21,. 

Art.  96  ;  pages  34,  35. 

1.    12x6  + 7t*+ 5a:3+ 10a:-4.  2.   -  ?«5  —  37  m2  +  70  ?«  -  50. 

3.  a5-5ai6+10a362-10n263+5a64_i5. 

4.  _6x5-25a:*  +  7x^J4-81x2+3.r-28. 

5.  8a7-44a5_40a*+76a3+ 112«2-,32.        6.   x^  +  y^  +  z""  -  3  xyz. 

7.  a262  4-  c^d'  -  d^c^  -  Jf-d'.       8.    9  x^  -  6  .r '  —  50 x^  +  60  x^  +  28  x  +  4. 

9.   x8  +  x*  +  l.  10.   x6- 50x1 +  769x2 -3600. 

11.    «6_ea452  +  9„2;,4_4;,6.     12.    ^2  -  4x^  +  4j/2  -  9^2.      13.   x^-/. 

14.    62-rf2  15    _4^s.  16.   0.  17.   80c.  18.   0. 

19.    -  9  m*  +  82  »i2?)-2  -  9  n*.  20.   8  a^. 

Art.  106  ;  page  40. 

1.    3x2  +  6x  +  9.  2.   x3-2.r2+x-2.  3.   024.306+5^,2. 

4.    »n3  _  ^2  _  14  ,„  4.  24.  5.   n^  -  2  an>  -6ab'^+7  b^. 

6.  x8-2x2-x+l.  7.    x+2y-Sz.  8.    a" -  6'«  +  C". 

9.  x3  +  2x2-x+l.  10.    2«2_oft  +  262.  11.    x+a. 

12.    (6+.c)o  +  6c.         13.    (x+^)-3.  14.    x  +  a. 


2  collegp:  algebra. 

15.    (m-ny  +  2(m-n)  +  l.     16.    x^'+ (a  -  b)x -ab.     IT.x'^-bx  +  c. 
18.    (n-3i).r+  (2«  +  5  6). 

Art.  117  ;  page  49. 

9.    a2_2a6  +  &2_c.2.     10.  .r*-.r2-2a- -  1.     11.  a:*  — 49.t2+ 84.r-3G. 

12.    a*-14a2  +  25. 

Art.  143  ;  page  65. 

7.    8.T-70.       8.   2x-5.      9.   3a  +  b.      10.   4a:2-6ar.      11.   3x-l. 
12.    2a +  36. 

Art.  152  ;  page  69. 

6.  x^-Wx^y  +  86  x^f  -  176  x^^  ^.  105  yi, 

7.  a-!  ^  2  a5  _  4  «^  -  7  a^  -  16 a^  +  32  a  -  8. 

8.  6  .j-s  -  31  x3  -  4  ar*  +  44  x3  +  7  .r^  _  10  x. 

9.  rS  +  3  .r*  -  23  x3  —  27  2-2  +  166  3;  —  120. 
10.    24  a6  _  64  a^Z)  —  58  a*62  +114  aSft-'  _  36  a^b*. 

Art.  170  ;  pages  82  to  84. 
,      0x2  — Or +4  ^     ^  ^     a2  +  i2  ^     x  +  ;/  +  z 


3x  +  2  a  ar  — 3/  +  z 

5  r+l  g      6m -n  ^_    3x-2  g     x2-3x,y  +  y2 

'    x2  +  X  +  1  '    5  H(  —  7  «  X  +  3  x2  —  xy  +  3  j/2 

11     2x-^--^"-^^-  i2..»'-^+12»>— 6m  +  6.  i3_    x  +  5. 

4      16x'^+12  42^3  x+i 

14     — ?— .  15.    -^^.  16.    -^^.  17.       •^'+^^+^     • 

.    x-2y  a  +  b  a2  +  62  (.,  +  1)  (.r?  _  1) 

18.    2:^-3^/.  19.    --ii!L.  20.    x2  +  x^.  21.    J':i^=l^. 

2x  +  3j/  7n  +  n  x  +  jy  —  2 

22.    0.         23.   ^^'"-")-         24.   -2i!£:^.        25.    -J^- 

(m  +  »)-  x^-a*  •  x2-_y2 

„p  -ot+ic  +  ra  2^  3  ^^j,      _2(x2—  1).._ 


(a  +  6)(/>  +  c)(c  +  «)  l  +  9x  x*  +  x2+l 


ANSWERS.  3 

Art.  186  ;  pages  92  to  94. 
1.    -3.      2.   91.      3.   -1.      4.   -7.      5.   8.      6.   1.      7.   5.      8.   17. 

9.   _ii.      10.   _?.      11.   31.      12.   3(a-l).     13.   ^^-      14.    — • 

'  5  "  a  +  i  a 

15.    „.        16.    1 17.    -n.      18.    -^^-        19.    .7.       20.   -.04. 

Art.  188 ;  pages  95  to  97. 
3.   29,  14.  4.   3  dollars,  48  dimes,  6  cents.  5.   82,  31. 

6.    A, 


am(n-\),  -3    a{n-\)  ^     IQQ («_;,)  „     17 


Hi  —  71  m  —  n  pt  23 

L.  55  min. 
13.    A,  12  miles  ;  B,  14  miles.  14.   283. 


9.    27A  min.  after  5.      10.   ?-=^,   2^^-      11.  55  min.     12.  $15,000. 


15.   First,  <E^Z^  ;  second,  ^ilizll.       I6.    5j\  or  38^^  min.  after  10. 
a  —  b  a  ^b 

17.  A,  3  days  ;  B,  4  days ;  C,  5  days.  18.   A,  $  750 ;  B,  $  500. 

19.    $  15.36.         20.   Greyhound,  72  leaps  ;  fox,  108  leaps. 
21.    Man,  84  cents;  boy,  42  cents.  22.    1  minute,  If |f  seconds. 

Art.  201 ;  pages  107  to  109. 
1.    ar,  18;i/,  6.      2.    a:,  .13  ;  y,  -  .2.      3.    x,-^;y,\\.     4.   a:,  13  ;  y,  3. 

5.    :r.-6;   y,  -  5.         6.    :,^ac{bm  +  dn)  .  yM{cn -am)^ 
ad  +  6c  ad  +  be 

n,:r,--,y,--        ».   x,  a  +  b;  y,  a-b.       9.   x,  (a+ 6)2;  y,  (rt-6)2. 

10.    X,  -;v,  --•        11.    a:,  — i— ;  V,  J7i  +  n.       12.  r, -2  ;  y,  3;  z, -4. 

'2^        4  wi  +  n     •" 

13.    r,  5;  3/, -2;  r,  2.  14.    x,  (a  -  6)2;  y,  (o  +  6)2;  2,  2a6. 

15.    X,  0 ;  (/,  -  5 ;  z,  3.  16.   x,  12  ;  y,  -  24 ;  z,  -  36. 

17.    u,  -2;  X,  3;  ,1/, -4;  z,  -  1. 

18.  .r,-(2a  +  36-4c);    y,  l(_2a  +  36 -4c)  ;    z,  ^(ia  +  Sb  +  4:c). 

19.  X,  a;  y,  a;  z,\.  20.    x,  a6c  ;  y,  a6  +  6c  +  ca  ;  ?,  o  +  6  +  c. 

Art.  202  ;  pages  llO  to  112. 

3     14  6  4.   — •  5.   60,  8.  6.   24  men,  12  women. 

19 

ad  +  6c 
2bd     '  '      2bd 


COLLEGE    ALGEBRA. 


8.   Number  of  persons,  ^"^ — ^^^ ;  each  received  'lAHL+Jll  dollars. 
bin  —  an  bm  —  an 

9.  542.         10.    A,  .$14;  B,  |44;  C,  $42;  D,  $32. 
11.    A,  ^^""P         ;  B,  ^"'"^*  •  (^  ^™"'' 


mn  +  np  — pm  —  n>)i  -\-  up  +  pm  uin  —  np  +  pm 

12.    A,  10;   B,  15.  13.    ^^"'  "  ""  dollars  at  ^^^(^-f>)  per  cent. 

m  —  n  bm  —  an 

14.    A,$117;B,$63;  C,.$36.     15.   Hind-wheel,  15 ft.;  fore-wheel,  10 ft- 
16.    Eate,  40  miles  an  hour;  distance,  112  miles. 

17.  A,  8  hours ;  B,  9  hours  ;  C,  12  hours.  18.   A,  6 ;  B,  5. 

Art.  227  ;   pages  128,  129. 

3.    x<A.         4.    a;<l.         5.    .r  >  If.         6.    x<2a.         7-.    x<a  +  b. 

S.    x>2,  y>4.        9.   .r  <  24,  //  >  3.        10.   :r  >  5,  a:  <  15.       11.   8. 

12.   19.  13.   32  or  33. 

Art.  233  ;    page  135. 

7.    £!!_i?  + 6 -—-}--.  9.    9a:6_12.r5-2xt+28.T3-15a-2_8a:+16. 

b^        b  a       a^ 

■  10.    x8  -  Sx^  +  28 -r"  -  56x5  +  70x*  -  56x3  ^  28x2  _  8x  +  1. 

11 .  8  x9  -F  60  x'  +  150 .7-5  +  125  x^. 

12.  64  a^xS  -  144  a^b^x^y  +  108  aVAry'^  -  27  b^yK 

14.  o^"'  +  15  a*""  6"  +  75  a^™  b-"'  +  125  b^». 

15.  xe-3x5-}-6x4-7.r3  +  6.r2_3x-f  1. 

16.  8.r6-.36x5+42x«-f9x3-21.r2-9x-l. 

18.  8  x9  -  36  x8  +  42  x^  -  39  x^  -f  123  x^  -  69  x*  +  23  x^  _  156  x2  -  48  x  -  64. 

Art.  268  ;    pages  162,  163. 
7.    2a2-5aZ.  +  8Z;2.  8.    l-x-f--  9.    2.r3  -  3x2 -f  4x- 5. 

10.    21.12.  11.    900.8.  12.    .8253.  13.    Sa'^ -'2ab  -  b^. 

14.    l-^-'^^  15.    x3-x2  +  x-l.  16.    31.7.  17.    10.13. 

3       2 

18.    .0534.  19.    .t2-2x-2.  20.    a^  -  b.  21.    r2-2x-2. 

22.    21.4.  23.    .40.  24.    12.3. 


ANSWERS.  5 

Art.  280  ;   page  170. 

1.73205.  5.   1.05409.  9.    1.25992.  13.  .72112. 

2.64575.  6.    .44721.  10.    1.81712.  14.  1.07722. 

.37947.  7.   .64550.  11.    1.93098.  15.  .63764. 

.04472.  8.    .42492.  12.    .31072.  16.  .87358. 


28.    243 


37. 


Art.  290 ;   page  177. 
29.    -128.  30.    ±243.  31.    1296. 

32.    6x2  — 7r3  — 19x3  +  5ar  + 9a:t  —  2xi. 
2x-^y-l0  xy-i  +  8  x^y-^.  34.    2-4  a~  t  xt  +  2  a~  I  x^. 

35.    ai-ah^  +  bi.  36.    a-i6-2  -  a-'-^^-s  -  a-^t"*. 

x-^-y-i  +  T^-y-l       38.  ah-^-2  +  a-h^.      39.  x^-3xi  +  2xi 
40.   a-i^V".  41.    x«-^  42.    a'^y.  43.   x. 


Art.  317  ;   pages  191  to  193. 
y/2.  2.    V3.  3.    ^5^.  4.    v/^.  5.    4^. 

7.   da^b^VUFc.        8.   3a6v'2a-56.        9.  (m-9n)V'W. 

V3. 


6a72, 

1 

6 


lV30.  11.    ^ 

9 


,   3a6v2a  — 56. 

12.    -^  ^18a-'6c2, 
4a-^ 


m 


13.    V448. 
\/25^.  15.    </6¥b^*.  16.    a/^-^^-  17.   2Vr^^. 

18.    v/625,   'C/216,   ''^^ig.  19.    v'^*6*,   v^65^,   '^c%3. 

^4<v^7<V'3.         21.   6\/2.         22.    ?^+-vl8.        23.  9\/3. 

4  3 


6aV3^.              25.    2Vx-^-y-^.               26.    2av/2a.  27.    VS. 

lla:_5-10Vx2-l.           29.    11-20a/15  +  5V10.  30.    '^32. 

'^^.        32.    3\/2.         33.    81  a^bx-^b^.          34.    ^7.  35.   v^^. 
>'86 


37. 

X 

40.    V^ 


</2^ 


38. 


024.  6  +  2rtV'6 


54-23\/6 


41 


2-6  43 

V6+  Vl5 


COLLEGE   ALGEBRA. 


43.   -17-8V2^-8v^-12\/2-12\/3-6>/2^9. 

gl  —  gibi  +  a2  bi  —  a%=  +  gt  4  —  g^ft  +  ahi  -  ab^  +  gU?  -  6? 

a^  —  6- 
7.24263....         46.    .10102....         47.    1.91245....  48.    Vl  +  Vb. 

5-2V2.        50.    \/5-V3.        51.   7  +  3V2.        52.   4V2-3V3. 
Vm  +  n  —  Vm  —  n.  54.    Va  +  .r  +  Va.  55.    v'3(7+\/5). 


v^(3V2-2\/3).      57.   -3. 

Art.  335;   pages  200,  201. 


K'^f'-c).    59.  12. 

g-36 


-2\/l5.        2.  -V^IT^.        3.    1.       4.    38_14\/^^.        5.  102. 
5 

-x^+2xy\/xy-y^         1.    -  10  +  9\/^^.  8.   0.  9.    7^2. 

V3.  11.   -4\/:r2.  12.    -2y/^Z.  13.    ~^^^~^. 


rt-2_6  +  2av'-6 


15. 


10  _  23  V- 10 


12 


a2  +  6  70 

19.    V-^  +  V^^. 
21.    Va  -  \/^^  22.    \/3  +  V- 


16.    12  V- 5. 
0.   5  +  V^. 


Art.  341 ;  pag-e  204. 
.    ±2\/in:.  3.    ±1.  4.    ±  Va  +  6.  5.    ±  \/l9. 

g  +  i 


2.      7.   ±3.      8.   ±2.      9 
2.  13.   ±1.  14 


(a -6).       10.    ±  V2.      11. 


15.   ±2. 


16.   ±  -  VlO, 


Art.  346  ;  page  209. 


17 


2.    3,  -  2. 

7.    •^.-*. 
4       9 

, ,     3         1 


2   1_ 
5   5 
1    1 
2' 3* 


12. 


4.    -^, -^. 

7       7 

-I+2V5 


-  5  ±  V-  23 


5.    15,  1. 

2 


4.   3, 


2 

2±  V7 


ANSWERS. 

Art.  347;   pages  210,  211. 


5.    .-1. 


10. 


-2. 
4       4 


5,2. 


2   3 


9 


12.    Il-     13. 

3  3 


■  39±  3\/ir3 


Art.  348;   pages  212,  213. 


-1,  -4. 

4     15   5           5     1     1 
4  '  2             ■    2'  14" 

6.13   1.          7.-1,-?. 

4                           2       2 

-"    -'• 

9.   5,  -3.              10.   4±2V3.              11.   4,  - -. 

- 10  ±  V/ 

■y. 

13.    1,~-       14.   2,1. 
36                   3 

15.4,-1       16.-^,-^. 

^■f 

18.   4,  ~.          19.    1,  -• 
4                       9 

20.  1^,-2.         21.    1,-18. 
23 

I-- 

23.   2.          24.    17,-1. 

25.   -2,  — •          26.   3,"^. 
65                        9 

^'-f 

28.   3,-2.          29.   5,- 

■  -•          30.   2V3-3,  2-V3. 

21±2vT09. 

32.1,-10.33.1- 

29      „.     8-5V2   13-9\/2 
21          ■          7       '         7 

Art.  349 ;   pages  215,  216, 


20. 


1 

_2 

lm-1.      4. 

04 

-.h. 

5.    a 

,-b.      6.   - 

-1, 

—  a 

.      7.   -16, -2c. 

m 

■:\- 

.3.  9. -^ 

a 

'  c 

10 

.   2p, 

-  5;j.      11. 

- 

h 

3n        -2    3^'   « 
'  2'          ■     4'2" 

n. 

n 

1  +  1 

14 

?n 

n 
111 

15.    a. 

1 
a 

16.    a +  6,  4  6. 

.1^- 

(a  +  ;.)^  - 

-(« 

-M 

-. 

18.    9m,  -I 

n. 

19.    a, -2a. 

a 

+  h 
c 

c 
a  +  h 

21.    , 

■i  +  b, 

2ab 
a  +  h 

22. 

w  —2n,  n  —  2  m. 

2 

—  a 

-2a-l. 

24. 

—  a, 

-b. 

25.  a-6,- 

■a- 

-c. 

23        6-2«  +  c_ 

c  +  a-26 

2 

ac 

6        3a  +  5 
6c 

il'. 

38.  « 

a 

+  6  rt  —  6 
-  6'  a  +  b 

29. 

3  n  +  4  6        a 
a       '  2a-\-  b 

30. 

a  +  6  —  c, 

a  — 

h  + 

c. 

31     -2a6±0 

2 

COLLEGE   ALGEBRA. 


S2    t±±±l,   «'-"  +  l.  33.     ^^^  +  "^)     - 6. 

a^  —  a  +  l     a-  +  a  +  l  a  (2  a +  3) 


34. 


5a  +  6      3rt  —46 
3a  — 26'    2«  +  56' 


35. 


a+6+c     a—h—c 


a  —  b  —  c     a  -{-b  +  c 


Art.  351 ;   pages  218  to  220. 


11.    0,  -1 


.    0,  3,  -8. 

l±  V^^ 


).    2,  -^,  -4. 

2 


10.    0, 


3        2 


12.    3, 


3±3V-3 


14.    +a,~,-~-      15.  -1,2,  ±3,  ±4.     16.    ±1 

'22  '2 


4        3 
13.   ±2,  ±2^^^. 


5     5±5V;^ 


io         4a     2a±2aV'^ 

^^-  -y       3 


19.    1,  ±3. 


21.    -5,  0,  ±2V3.  24.    -a,  -b. 

3 


25.    0,  =1±:^.       26.    0,  - 1, 


1  ±  V-3 
2 


2        5 


J.  ±1,  ±2. 


0,  +5V2.  30.    ±2a,  ±3a.  31.    1,  —4, 


5±  >/41 


0,  ±  Vl3.  33.   0,  ±  Va^  +  6-^. 

3.  II.  36.    a,  b,  IL+i. 


34.    1,2,3, 


11  ±  V-23 


37.    -1  ±  V5,  i2V-l. 


1     _2 
2'        3" 


Art.  354;   page  224. 

2.    a:2-9r=-20.  3.    .r2+2.r  =  3.  4.    5r2_12.r=9. 

5.    3x2  +  40ar  =  -133.       6.    12.r2_  17r  =  -  0.       7.  21x2  + 44a:  =  32. 

8.    33-2+11x1^0.  9.    G.r2  +  31.r  =  -35. 


10.    x2  _  (2  «  +  6)  X  =  -  «2  -  ,t6  +  2  6- 
12.    a:2-4x=71. 


11.   .r2 -2 //(.(•  = 
13.    4  x-  _  4  11,  f  =  H  —  «j2. 


ANSWERS.  9 

Art.  355;   pages  226,  227. 

6.    (2x-Z){x+5).         7.    (5x+l)(x+7).         8.    (2:c-3)(4a:-3). 

9.    (13  +  x)(3-a-).  10.    (l  +  2.r)(2-3a). 

11.    (x  +  2  4-V3)(^  +  2- V3).         12.    (3x-H-  V5)(3x-1- Vo). 

13.    (6:r-5)(2x+l).      14.  (2  +  x)(3  -  2.0-      15.   (4.c  -  l)(2.r  +  5). 

16.  (.r-2)(10z-3).  22.    (Sx- 7  7«n)(7x- 3//)?0- 

17.  (Vl7  +  4  +  x)(Vl7-4-.r).  24.    (x  +  Sy -2)(x  -  2 y  +  3). 

18.  (6+12.i)(3-2x).  25.    (x  +  y  +  l)ix  +  2y  +  2). 

19.  (2.T-7a)(3z+5a).  26.    (2  -  3^  +  x)(3  +  2  */ -  a). 

20.  {dx  +  4:m)(4x  +  5w).  27.    (x- ;/ +  2)(2r  +  j/ -  1). 

21.  (3x-2(/)(4x  +  5(/).  28.    (3fl  +  6  -  4)(a  +  6+ 3). 

29.     (x-3//-s)(x-2j/-4s). 

Art.  356;   page  228. 
5.    (x-3)(4x  +  3).        6.    (2x-3)(8x-5).        7.    (3x-2)(x  +  3). 

8.    (3x-2+ V3)(3x-2- V3).      9.  (6x- l)(Gx  +  5). 

10.    (4x  +  5)(2x+7).     11.   (G  +  5x)(7-2x).     12.  (5x-3)(3x-l). 

13.    (5x-2+ V'6)(5x-2-\/6). 

Art.  357  ;   pages  229,  230. 

4.    (x2  +  x+l)(x2-x+l).  5.    (x2  +  3x+l)(x2_3x+l). 

6.  (2a2  +  2a6-62)(2a2-2a6-62). 

7.  (?«2  +  3mn  +  5n2)(m2  — 3?nn  +  5n2). 

8.  (l  +  3&-262)(l_36-262). 

9.  (x2  +  4x^  +  2«/2)(x2-4xy  +  2!/2). 

10.  (2a2  +  2a  +  3)(2a2_2a  +  3). 

11.  (2  w2  +Sm  +  4)  (2  m2  _  3  w  +  4) . 

12.  (a2  +  ax  V3  -  x2)  (o2  _  ax  VS  -  x2) . 

13.  (x2  +  3xV2  +  9)(x2-3xV2  +  9). 

14.  (2a2  +  a&  +  462)(2a2-a6  +  462). 

15.  (4  .r2  +  5  mx  —  3  m^)  (4  x2  —  5  mx  —  3  ni^) . 

16.  (3.r2  +  3xV2+2)(3x2_3xV2  +  2). 

17.  (3a2  +  4o,n  +  5  Jn2)(3a2  — 4am  +  5w2). 

18.  (2  +  2n-7«2)(2-2H-7n2). 

19.  (4  x2  +  3  xy  -  5  y2)  (4  x^  _  3  xy  -  5 ^2) . 


10  COLLEGE   ALGEBRA. 

Art.  358;   page  230. 

1.    \/2  ±  V^,  -  V2  ±  V=^.  2.  1  ±  \^,  -  1  ±  V2. 

5.    1  ±  V3,  -  1  ±  V3.  6     ^^^^^^^    -Vl4±\^r2 


Art.  362  ;   pages  235  to  238. 
3.    19,8;  or, -^,-^.  4.3,4,5.  5.   4,  16;  or,  -  5,  25. 

6.    8,  5.         7.    1,  2,  3,  4 ;  or,  5,  6,  7,  8.         8.    16  barrels,  at  $  6  apiece. 
9.    §  30.  10.    4,  5,  6 ;  or,  -4,-6,-6.  11.61  miles  an  hour. 

12.  27  and  36  miles  an  hour.        13.    9  c/.  per  dozen.        14.    $80  or  $20. 

15.    20.  16.    9  miles  an  hour. 

17.  Area  of  court,  529  sq.  yds. ;  width  of  walk,  4  yds. 

18.  Hind-wheel,  16  feet ;  fore-wheel,  12  feet. 

19.  Larger,  5  hrs. ;  smaller,  7  hrs.  20.24.  21.  $2000.  22.8.  23.5. 
24.  4.  25.  6.  26.  14400  and  625  sq.  ft. ;  or,  8464  and  6561  sq.  ft. 
27.    12.  28.   38  or  266  miles.  29.   4,  -  5.  30.   70  miles. 

31.    A,  18;  B,  12.  32.    100  shares  at  $15  each.  33.   42. 

Art.  364;   page  241. 
4.    ±4,  ±3.       5.   ?, -2.      6.    ±3  ±2        7     ?  -1      8    -^ 1_ 

9.    ±lV3,  ±1.  10.    -1,  -i-  11.    vl2T.  4.  12.    25,    ^. 

^3  2  '16 

13.  -243,   v'265.  ■"■     '•  •^^"     '      ^^^- 


u.  (..,.,  {-ff 


■".    le,  (P?-        n.    I  (If-        18^    .l(-r\^.       19.    -32.1,^2 


4   \'6J  8'\     74y  ■         "'  4 


Art.  365;   pages  244,  245. 


3.    5,-l,±3.     7.1,3,-5,-7.     8.  ±  2,  +  ^30      9    2       ^   3^a/505 

2  2'         4 

10.-1,-2,^,^.         ll.±3,±3V2.         l2.-l,-2,±^. 
3   3  2 


13.    1, 


15 


14.    3 


17.    2,  3,  1,    I- 
2     3 


ANSWP:ilS. 

.-21.  15.6 

13 

18.   -Sf-lO, 


11 

-9.  16.   4, -6,  ±1. 

1  ±  V-  4679 


19.    0,  -2,  -1±2V-1. 

22    i,   -1,    ^^^^.         23 
3        3  50 

25.   -2,-3,-4,-5. 

5  ±  V2T 


-2,  -VIS.        21 
1     3±  V;^503 


a±Va2_462 


24.   1,  1±2\/15. 


27.    -1, 

39.    7,  1 


31. 


0,  2,  -  7, 
.        19 


3.   0,  2  +  2a±2V\  +  2a. 

29.    2,  1,   9W=31. 
4  8 


3±  a/5 


rl  7,t 


33.    a +  65,  a  +  Sn^.        34.  1,  2, 
1         o         2 


7     _10 
3'        3' 


35.    3.  -1 


2±  V61 


36.   t,    -2,  - 
5'  11 


37. 


5±  V2l     -5±\/2T 


^  V7,  ±  —  v'-  665. 


Art.  367  ;   pages  247,  248. 

Note.  In  this  and  the  four  following  articles,  the  answers  are 
arranged  in  the  order  in  wliich  they  are  to  be  taken ;  thus,  in  Ex.  2, 
the  value  a-  =  3  is  to  be  taken  with  y  =  +  5,  and  x=  —  S  with  y  =  ±  5. 


,77  99 

3.    a:,      or ;  w,  ±  -  or  ±  — 

'2  2  ^'      2  2 


2.   a-,  3  or  —  3 ;  y ,  ±  5  or  ±  5. 

4.  ar,  2V3or-2\/3;  y,  ±2\/2  or  ±2V2. 

5.  x,2a  —  boT  —  2a  +  b;y,±(a  +  2b)ov±  (a +  26). 

6.    a-,  -or--;  y,  ±-or  +-•  7.    ar,  3  or  -  3  ;  y,  ±  2  or  ±  2. 

'2  2'^'      3  3  '  ^' 


r,  1  or ;  w,  —  2  or 

25     ^  25 

T,  6  or  -  9 ;  y,  -  9  or  6. 


Art.  368;   page  249. 

46  4.    .r,  8  or  —  7  ;  y,  7  or  —  i 


X,  10  or  —  3  ;  y,  17  or  4. 
T,  2  or  —  5 ;  y,  5,  or  —  2. 


COLLEGE   ALGEBRA. 


x,6ot1;  y,l  or  6. 

x,a-\-l  OT  —a  ;  y,a  or  —  a  — 1. 

X,  4;  y,  6. 


1 

2 

12.    a;,  3  or  —  ;  y,  2  or  -  — • 
3      -^  3 


10.    X,  a  ±b;  y,  a  +  h 


14.    :r,  2«-3  or 


10a 
13 


13.    X,  4  or 


y,  3a  —  2  or 


12 


12 
7  '  ■"  7* 

126  a -169 


Art.  369  ;   pages  251,  252. 

4.  r,  8  or  6  ;  !/,  6  or  8.  10.  x,  -  1  or  -  8 ;  ^,  -  8  or  -  1 

5.  X,  1  or  -  11 ;  y,-\\  or  1.  11.  x,  2  or  -  16 ;  y,  16  or  -  2. 

6.  x,  —  1  or  —  4 ;  ?/,  4  or  1.  12.  x,  8  or  —  2  ;  y,  —  2  or  8. 

7.  X,  8  or  3 ;  y,  -  3  or  -  8.  13.  x,  17  or  4 ;  j/,  -  4  or  -  17. 

8.  X,  ±  3  or  ±  2 ;  y,  ±  2  or  ±  3.  14.  x,  ±  11  or  ±  8 ;  y,  +  8  or  +  11 

9.  X,  5  or  -  3 ;  !/,  3  or  -  5.  15.  x,  2  or  -  10 ;  y,-  10  or  2. 

16.  X,  a  +  h  ;  y,  a  ^  b. 

17.  X,  3a +  2  or  2a- 3;  y,  2«- 3  or  3a +  2. 

18.  X,  -  6  or  -  25 ;  y,  25  or  6. 

19.  X,  a  4-  1  or  —  a  ;   y,  a  or  —  a  —  1. 


Art.  370;    page  253. 

2.  X,  ±  4  or  ±  -  V2 ;   y,  ±  1  or  +  -  y/2. 

3.  X,  ±  3  or  ±  -  \/3 ;  v,  +  1  or  +  i  Vs. 

3         ^  3 

4.  X,  ±  2  or  ±  -  \/5  ;   V,  +  3  or  ±  -  Vs. 

5  -^  5 

5.  X,  ±  4  or  ±1 ;  y,  =f  2  or  +  3. 

6.  X,  ±  4  or  ±  ?  V?  ;  !/,  ±  3  01  T  -  v^. 

7         ^  7 

7.  X,  ±  2  or  ±  A  \/-13;  y,  +  1  or  ±  A 

13  -^  13 

8.  -r,  ±5or  il6;  y,  +  1  or  + -• 


1  2  or  ±  —  V3 ;  y,  ±  5  or  +  —  V3. 
15  15 

1  o  3  9 

±  1  or  ±  2 ;  V,  ±  -  or  ±  — 

'  •"     -1  8 


ANSWERS. 


13 


Art.  371 ;   pages  256  to  258. 


6.  .r,  —  or  -  2  ;  y,  -  or  -  5. 

11  •^    11 

7.  X,  3,  2,  or  -  3  ±  VS  ;  y,  2,  3,  or  -  3  +  VS 

8.  :r,  ±  2  or  ±  -  a/5  :  y,  +  3  or  +  -  V'5. 

5        '  -^  5 

9.  z,  ±2  or  ±  14;  y,  +3  or  t5. 


10.   X, 

12.    X, 


11.    x,^  OT-3;  ij,±-OT  ±V3. 
'2  '  -"     2 

4,  -,  8,  or  -  -;  y,  3,  8,  -5,  or  16.      13.   x,  1  or  1;  y,  -1  or  -  1. 


1111 

-  or  -  :   (/,      or  — 
2       9     -"    9       2 

3 


14.    X,  2,  1,  or 


1,-2,  or 


7       4-  "'      4 

-  3  ±  V^^SS 


15.  X,  3, -6,  1,  or  --;  y,  -  1,  1,  - -,  or  -• 

2  2^  4         4 

16.  X,  a  —  b  or  b  —  a;  y,  a  +  b  or  2  a. 
1111 


1  1 

4       3-^0       4 


17.    X, 

19.    X,  2  or  16 ;  y,  2  or 


18.    X,  ±  a  +  b;    y,  ±  a  —  b. 
±3  or  ±2;   y,  t  2  or  +'3. 


21.  X,  ±  2  or  ±  i  VT  ;  y,  ±  5  or  +  —  V?. 

7        ^  7 

22.  X, +  (2a-6)  or  +(a-2b);  y,±{a-2b)  or  ±(2 a  — 6). 

23.  X.  2,  1,  or  3-AV=X9  ;  ,,  i.  2,  or  ^  ^  ^^. 


10 


6±V'193  .       „,         63±3vi;93 

•  y,  —  5,  —21,  or 


'    '       3  4        -  -         -  4 

343  or  -  125  ;  y,  125  or  -  343.  26.   x,  2  or  -  1 ;  y,  -  1  or  2. 


27.  X, 
29.  X, 
31.    X, 


2  or ;  w,  —  1  or  2. 


27       27        ^        9  9        ^ 

X,  — , ,  or  0:  ?/,  -,  -,  or  0. 

'2         4  -^   2  4 


3  or  —  7 ;  y,  —  1  or  —  21.  30.    x,  2  a  or  —  a  ;  y,  2  i  or  -.-  b. 

2,  0,  or  ±  \/2 ;  y,  2,  0,  or  2  +  V2.  32.    x,  3 ;  y,  -  2. 


33.   X,  ±  XflliJ:  or  0 ;  y,  ±  Va'^  +  1  or  0. 


34.   X,  ±  3  or  ±  V-  7  ;  y,  2  m-  6 

36.  X,  —  1  or  —  1  ±  VlO ;  y,  -  1  or  9  +  2  VTO, 

37.  X,  2,  1,  6,  or  -  3;  y,  1,  2,  -  3,  or  6. 


35.    X,  ±  1  or  ±  —  :  v,  +  -  or  ±  — 
59^       3  31 


14 


COLLEGE    ALGEBRA. 


38.  a:,  3,  -  1,  -  1,  or  -  2 ;  y,  1,  -  3,  2,  or  1. 

39.  X,  3,  4,  or  —  6  ±  V43 ;  y,  _  4,  -  3,  or  6  ±  ViS. 

40.   X,  8, 2,  or  2  ±  4  V^^;  y,  2,  8,  or  2  +  4\/^^.     41.  a:,  2  or  1 ;  i^,  1  or  2. 

43.   ar,  4  or  —  5 ;  y,  —  5  or  4. 


42.   a:,  ±  -  or  ±  - ;  y,  ±  1  or  +  — 

'2  7^  19 


44. 


1  or  —  ;  V,  2  or  -  —. 
20     -^ 


60.    x,±^-y/2;  y,±--\/2: 
'6        '  ^'     2 


V2. 


Art.  372  ;   pages  259  to  261. 

1.    ±  9  and  ±  5 ;  or,  ±  V53  and  ±  v'53. 

2.  8and±3;-8and±3;  3\/^and±8\/^;  or,-3\/^and±8 V^. 

3.  Length,  10  rods ;  breadth,  6  rods.  4.   4,  7.  5.   8,  5. 

3         -U5 


14 


6.    Duck,  #1.75;  turkey,  $2.25.    7.  21  or  12.    8.4,9.     9. 
10.    Length,  150  ft. ;  breadth,  100  ft. 

11.  Length  16  rds.,  breadth  10  rds. ;  or,  length  13V  rds.,  breadth  12  rds 

12.  Rate  in  still  water,  4  miles  an  liour ;  rate  of  stream,  2  miles  an  hour. 

13.  $3.  14.    A,  6  hrs.;  B,  3  hrs. ;  C,  2  hrs. 

15.    Distance,  504  miles;  A,  24  miles  a  day;  B,  18  miles  a  day. 


16. 

7  and  5 ; 

1.3+V^^^^-1,3-V-71. 
2                                  2 

17. 

5  and  2  ; 

^^^7+V-3O3^^^,7-V-303. 

18.    3  and  1  ;  -  1  and  -  3 ;    1  +  V^  and  -  1  +  V^  ;    or,  1  -  V^ 
and  —  1  —  V—  G. 

19.    3  miles  an  hour.  20.    -150  miles. 


ANSWERS.  15 

21.  A,  8  miles  an  hour ;  B,  6  miles  an  hour. 

22.  A,  11  hrs. ;  B,  22  hrs. ;  C,  33  hrs. 

Art.  375  ;   pages  266,  267. 

4.    x=9,  y  =  l;  x  =  6,  y  =  S;  or,  x  =  3,  ij  =  5. 
5.    X  =  4,  y  =  13  ;  or,  X  =  8,  »/  =  6.  6.   x  =  3,  y  =  5. 

7.    x  =  4,  y  =  122;  x  =  13,  y  =  9\;  x=22,  y  =  60;  or,  x  =  31,  y  =  29. 

8.    X  =  3,  2/  =  50  ;  X  =  10,  y  =  26  ;  or,  X  =  17,  y  =  2. 
9.    X  =  3,  y  =  2.  10.  X  =  3,  y  =  89  ;  or,  X  =  14,  y  =  43. 

11.  x=78,  !/  =  4;    x=59,  y=12;    x  =  40,  y  =  20  ;    x  =  21,   y  =  28; 

or,  X  =  2,  y  =  36. 

12.  x=r2,  3/=l.  13.    x=5,  j/  =  2.  14.  x  =  8,  y  =  6, 
15.    x=3,  y=ll.                  16.    x=7,.y=l.  17.  x  =  9,  y  =  4. 

18.    X  =  2,  y  =  l,  z  =  S. 
19.   X  =  2,  y  =  31,  2=3;  or,  x  =  13,  y  =  l,  z  =  3i. 

20.  Either  1  and  17,  3  and  12,  5  and  7,  or  7  and  2,  fifty  and  twenty  cent 

pieces. 

21.  Either  6  and  3,  or  1  and  7,  florins  and  half-crowns. 

22.  1^  and  ?;  1^  and  I ;  or,  ^  and  !?• 

9  5      9  5  9  5 

23.  Either  1,  18,  and  1;  4,  10,  and  6;  or,  7,  2,  and  11,  half-dollars, 

quarter-dollars,  and  dimes. 

24.  5  pigs,  10  sheep,  and  15  calves. 

25.  Either  50,  2,  and  33 ;  28,  21,  and  50 ;  or,  6,  40,  and  67,  half-crowns, 

florins,  and  shillings. 

Art.  398;   pages  276  to  278. 

5.     9.  6.    25.  7.    5.  8     a-S^  9     21.  ^^     ^_3 

35  27  3  a  +  3  2 

11.    Srtjui.  12.    3,  --•  13.   2,-3,^.  14.    ±~. 

2  11  9  b 

15.    X,  ±a26;  2/,  +62a.        16.    x,  32;  y,  18.        17.   25,  11.        18.  31,  17. 
19.    6,8.         20.   9,3.         21.    ^^,    ^i^-        26.    3:4.  27.  «:-6. 

28.    ^  =  -JL^  =  ^-     29.    -  =  ^  =  i-    30.  A,§105;  B,$189;  C,.$270 
17      -23      -27  a      b      c 


!«)  COLLEGE   AL(;EBRA. 

31.   First,  wine  :  water  =1:2;  second,  wine  :  water  =  2:1.      32.   5  :  4. 
33.    6,  4,  12,  8  ;  or,  44,  -110,  -64,  160. 

Art.  413;   pages  283,  284. 


3. 

6.          4.   3/  =  |.2.          5.    70. 

6.    14.          7.   — • 
10 

8. 

J-V,. 

9. 

788-rVft.                   10.   5x,  -1 

X 

n.  f 

-¥■ 

13. 

y=S  +  5x-4x^.          14.    9. 

15.   6  in.           16.   3. 

17.   10. 

18.   8.  19.    15(\/3-l)  in. 


Art.  418 ;   page  286. 


2.    I,  147  ;  ;V,  1425.          3.   /,  -  131 ;  S,  - 1496.         4.   I,  36  ;  S,  -  264. 

5.   l,-^;  S,-78.           6.   /,  229.  ^,2925.            7    /  61          3439_ 

4                                        4  4                      '  2  '     '     6 

8.   Z,  -  —  ;  S,-  165.  9.   /,  -  6  ;  ,S',  -  — • 

4  5 

10.   /,  420+196;  6',  198a +  636.  11.    /    17// -8a:      o  80,y-35x^ 

2         '      '  2 


Art.  419;   pages  289  to  291, 
4.    a,  1  ;  ,9,  540.  5.    a,  7  ;  /,  -69.  6.   d,3;  S,  552. 

7.    f/, -5;  7,-145.  8.   n,  35  ;  d,--  9.   a,§:  (^,_J-. 

4  5  15 

10.  /,~;n,  16.      11.  n,  22;^,i.      12.  a, -3;/,  5.      13.  a,  -  -  ;  n,  14. 
12  2  5  .. 

14.    a,l;d,-l.  15.    n,  43  ;  J,  -  1.  16.   Z, -12;rf,_l 

2  3  3        '     3 

17.    a  =  -l,  n=16;  or,  a  =  — ,  n  =  25.  18.    n,  15  ;  /,  — • 

3  15  .4 


21     ^^2.9-fm.  ^^^2(.S-a«) 
«         '  n(n-l) 


22.  a_2^-»(n-l)«?.  ^  _  2  .9+ »  (n  -  1)  t/ 

2«  '  2h 

23.  n^^-"  +  ^-  ^._(/+«)(/-a  +  </) 

(^         '  2d 


ANSWERS.  17 


?_(„_!)(/;    ^=^{2/-(n-l)rf}. 


25     g^2^-n^.  j^2(nl-S)  gg     ^^- rf  ±  V8c/6' +(2« -c/)^ 

n        '  n  (n  -  1)  2 


27.    n 


a+/  2^- 


^^d±  V(2l+dy^-SdS.    ^  ^21  +  d  ^  VC2l  +  dy^-SdS 
2  '  2(? 

Art.  422  ;   pages  291,  292. 

d^--        2.    d  =  ---        3.    d=^-        ^.    d  =  -—.        5.    d=--. 
7  5  4  12  4 

am  +  b  Inn  +  a.  ^     ^  g     J^.  g     g  a6.  10.    "'  +  ^\ 

HJ  +  1     7n  +  1  15  a^  —  U^ 

Art.  423 ;   pages  293,  294. 
62750.  4.   -43.  5.    10,2,-6,-14.  6.   65  a +  52  6. 


7. 

'^"^"  +  ^^-                 8.   3,  5,  7,  9.                 9.   44550.                10.   31. 
2 

-  4,  -  1,  2,  5,  8 ;  or,  — ,  — ,  — ,  -^,  --•                       12.    4117i  ft. 

11. 

13. 

3,  7,  11 ;  or,  ^,  ^^  ^1.           14    30.           15.    mp  -  nq ^          ^g     ^^ 

'     '       '         2    2 '   2                                                »n  -  n 

17. 

15.       18.  -  3,  7,  17  ;  or,  -  3,  _  ^,  -I^.        19.  $2950.       20.  852. 

21.    3,  5,  7,  and  4,  5,  6 ;  or,  21,  5,  -  11 ,  and  22,  5,  -  12. 

Art.  427 ;   pages  296,  297. 

8.   /,6561;  ^,9841.       4.   /,  ^;  S,'^^-       5.   /, -1250;  ^,-1042. 
'      '  '243  243 


i.    Z,    ^   ;.9,255.       7./,      1280;^,       ^^l^-       8.  /,  -  ^^ 
128           128                                             2                        125 

-•5' 

,243             46.3.       J,        243           2343.       ^^^ 
64               192                    1024           1024 

...f^. 

Art.  428;   pages  298,  299. 

t.    a,  1;  5,611.                    4.   r, -4;  <S,  1G38.                   5.    a. 

^^'•a>- 

^•"■"'^•'•il-             '■■'•^S.L- 

18 

COLLEGE   ALGEBRA. 

8. 

,-5.^-19171                     §,^.4039. 
2            384                       2            384 

9.   r,l;„,9. 

10. 

^'~324'"'^'                      11.    a,  3;  n,  7.                     12.    r,  1 ;  n,  10- 

13. 

1     «+('--l)'^.              14.   r-'^-«.             15.   a- 

r                                        S-l 

rl-(r-l)S. 

16. 

a-     '-,8-     ^(f-^).         n-g-'^C-^);/- 

Sr'^-Hr-1) 

r"-i              r«-i(r-l)                             r"  - 1 

,-»_l 

18     ,      "-1/7.    o      /"~-a^ 

18.    r         ^        6        ^        _^. 

Art.  429  ;   page  300. 

2. 

9.    3.    'A    4.    -^     5.    -5.    6.    14.     7.    12.     8.    - 

2             5                 6                              5             55 

Art.  430;   page  301. 

9       Q     64 
'io'     ''•    147" 

2. 

8            3     11           4    25           g     581            g      916 

y     2284 

11             '27             '36             ■    990             ■    8326 

2475* 

Art.  433;   page  302. 

1. 

7- =  2.          2.    r=-3.         3.    r=±2.         4.    r=±5. 

5.    r=-4. 

6. 

r=±?.          7.   "'+^^^.            8.    5.            9.   4x2-9(/2.          lo.    ?. 
3                                                                                                         6 

Art.  434;  pages  303,  304. 

2.  3.       3.  5,  10,  20,  ^0 ;  or,  -  15,  30,  -  60,  120. 

4.  5,  15,  45 ;  or,  40,  -  20,  10.         5.  ±  4.         6.  3100  ft, 

7.  2,  4,  8,  16;  or,  810  _ 540  300  _240,    g_  _81_. 
13    13  13'   13       8192 
9.  3,  9,  27;  or,  25, -35,  49.  10.  8,  4,  2  ;  or,  -  8,  4,  ^2. 

11.  -3,4,  11;  or,  13,4,  -5. 
12.  4,  2,  1.    13.  A,  $108;  B,  §144;  C,  $192;  D,  §256. 
14.  -4,  1,6,  30;  or,  8,  1, -6,  36.      15.  3.      16.  a,6;r,  ±2. 
5700  1710  513 
■'  139'  139'  139' 


17.  12,  18.  27  ;  or.  5^.  i^,  *^^-^        18.  ""I^/^ 


ANSWERS.  19 

Art.  441 ;   pages  307,  308. 

3.    A.       4.    -A.         5.    ^.         6.    -A.  7.    -A.       11.    _72. 

74                  19                169                   29  142                    5 

12.    5?.:^.                   13.    . ^ -.  14.      "^("  +  1)  . 

a2  +  62                              na—nb  —  a  +  2b  bm  +  2a  —  b 

,r          XV              x//                xi/                ,«    o         7  ,„           3 


16.  3,---        17.    -— •       18.  24, 


2x-y3x-2y   ix  —  Sy  2  19 

19.    1  1  1-  or  1  i   I  20.   MzlLziiO. 

2   5  8  8  5  2  mq  —  np 

Art.  446  ;   pages  313,  314. 

9.    zS""  -  10  x*™  j/2"  +  40  x^'"  y4™  -  80  x-^""  (/C»  +  80  x"  ?/8«  -  32  ^/W". 


12.  x-W  _  ^  x-8  v3  +  -  X-6  y6  _  12  x-4  y^  +  —  X-2  V^^  -  —  lA^ 

3        ^9         ^       27  81        •^         243 -^ 

13.  J  b-  ¥  +  7  J  b-  ¥  +  21  at  6-2  +  35  a5  6-  S  +  35  a-  2  6t  +  21  a"!  62 

+  7a-t6¥  +  o-2-6¥. 

14.  o«  +  16  a¥  +  96  a V  +  256  a '^  +  256  aJ. 

15.  32.r5^-t  —  40xty-i  +  20x^yi  -bxiy^  +  -  x-^yi  -  —  x~^y^. 

8  32 

16.  a-18  -  2  a-15  xi  +  -  a-12  x  -  —  a'^xt  +  A  a-6x2  -  A  a-3;j.t  +  —  x^. 

3  27  27  81  729 

17.  x3  +  15  x"V- !/- 1  +  90  xt  j/"  i  +  270  xt  ?/"  t  +  405  xt  ^- 1  +  243  y-^. 

18.  81  a-362  _  108  a-2  6-2  +  54  a-^-^  -  12  6-w  +  a6-i*. 

19.  a36-3  +  12  a26-2  +  60  a6-i  +  160  +  240  a-^b  +  192  a-262  +  64  a-W. 

21.  x6  +  3x5+6x* +  7x3+6x2+3x+ 1. 

22.  27  ac  -  27  a^x  -  99  a4x2  +  71  a^x^  +  132  a^x^  -  48  ax^  -  64  x^. 

23.  1  +8x  +  20x2  +  8x3-26x4-8x5  +  20x6_8x7  +  x8. 

24.  10  xi2  +  32  xio  +  96  x9  +  24  x^  +  144  x'^  +  224  x«  +  72  x^  +  217  x* 

+  228  x3  +  54  x2  +  108  x  +  81. 

25.  1  _  5x  +  15  x2  -  30x3  +  45x4  _  51  ^^  +  45 x«  - 30 x^  +15x8-5x9 

+  x"'. 

26.  .ri"  +  10  x^  +  30 xs  -  120  x«  -  48 x^  +  240 x*  -  240 x^  +  100  x  -  32. 


20  COLLEGE    ALGEBRA. 

Art.  447  ;   page  315. 


2. 

462a5a-6.       3.   ^368  vr\        4.    -792^/",        5.    SOGOaS.        6.    84 ''^ 

6m  n +9 

7.    -5005x    "     .             8.    ^a-%'\              9.   219648 a;-6(/i 

10.  - 486486 a-3.r25.         H.    126720.                12.    -^z"a?\ 

13.    110565.riV.              14.    -i^.r". 

Art.  468 ;   page  327. 

1. 

2-ll.r+33.T2-99.r3+-...             2.    1  _  4.t  +  20.r2  -  92.r3  +  ... 

3. 

24^8          16                                  3      3^9          3 

5. 

1_.,_1,-2_1,3+....                      6.   «_26-3^-3^+.... 
2          2                                                                2  a          a2 

7. 

^2:r2      16  .r6  ^  64  .rio  ^                              3          9      ^  81       ^ 

9.    2«2_ll-l-^_Ai!i_....        11.    Convergent. 

4  a"      32  aW      768  aie 
12.    Convergent.      13.  Convergent.     14.  Convergent.     15.  Convergent, 
16.  Convergent.  17.  Divergent. 

Art.  471;   pages  331,  332. 

2.  l-2.r  +  2.r2-2x3  +  2.ri 

3.  2  +  11  .T  +  33x2  +  99^.3  ^  297 .r*  +  ...  . 

4.  3-19x2  +  95.r*-475.r6  +  2375x8 

..2      ,  4    ,  ,    8    5  ,   16    -  ,    32     „  , 

3        9         27          81          243 
6.    l-2x+2x3-2.c*  +  2.r6+....     7.  x-x^-2x^-5xi-l2x^ 

8.    2-X+ 3x2-x3  +  3.r*+ ■■■  •     9.  1 -2.r  +  5.x-2- 16.t3+ 47x1+ ..- 
10.    2  -  7  x  +  28  x2  -  91  x3  +  322  x*  +  .  •  •  • 


12. 

l  +  lx  +  1..2_§..3_ll.,5+.... 

3        3         9         27 

13. 

2      4         8          16     ^32      ^ 

15. 

2  _„  ,  8   _i  ,  32  ,  128      ,  512    , 

-  X  2  4-  _  .c  1 4 ^ x-\ t2 

3  9            27       81          2 13 

16. 

x-J  +  3  +  2x'-5x2-16.,' 

ANSWERS.  21 


17.    a;-2-.r-i-2.r  +  2x2-4x3+  •••  • 
la     3   _3      1    _2      1    _i  ,   31  ,  23      , 

2  4  8  16      32     ^ 


Art.  472  ;   page  333. 

2.     l  +  .r_ix2+l.r3-^x-i+  ...  . 

2         2  8 

2^      8"^       IG''        128  "^ 

4.  1- 

2 

5.  l+l:r-^:r2+-:r3-ilx4+  ... 

2  8  16  128 

6.  l-^x-'^x^-  —  x^-  —  xi 

3  9  81  243 

7.  1+1..  +  2,._L3,3.+  A,4+... 

3        9         81  243 


Art.  474;   page  335. 

4.    ^ i—.  5.     ^  1  1 


2.r  +  5      2.r-5  .t      3x+5  2x      x+S      x-S 

2  I        3        ^        2 ^       3a  2a  g  1        ,       2 

T      3x+l      2.1--5  ■    x+a      x-ia  '    3  +  ix      3~x 

-J_+^ ^.  10.   -1 ^ ^  +  -1-. 

x+l      2:f+3      2z-3  .r+2      .r-2      x+1      x-l 

11.  1+  V2        ^        1-V2 


2X-5+V2      2X-5-V2 


Art.  476;   page  337. 


2 

2 

x+5        (:» 

6 

23 

3 

•  +  5)2 
1 

,r+l 

1 

(x+iy 

(.r+l)3 
4                , 

14  4 


-2      (x-2)2      (x-2)3 
2  4  3 


3x+2      (3a- +  2)2      (3.r  +  2)3 
1  1  1 


5(5ar-2)      5(5a;-2)3  x  +  1      (z  +  1)2      (x  +  1)3      (r+l)^ 

8.  -2 i__.+  _3 1_. 

x-l         (x-l)2         (x-l)3        (x-1)* 

9.    1 27 27 

2(2x-3)      2(2  X- 3)3      (2x-3)* 


22  COLLEGE   ALGEBRA. 

Art.  477  ;   page  339. 
„23  5  ,515 


X      x+2      (x+2)^  X      x^      x+l      (x +!)■'= 

-1—         1  + § 5.1  +  -^+^-  +  -^— 

x-2      2(2x-3)      2(2x-3)2  x      x-1      x-2      (x  -  2)2 

1_2        3^_^L_  7     5_1,2 5  4 

X        X2  x3        X  +  5'  ■     X        X2        x3        X+1         (x  +  1)2 

Art.  478 ;   page  340. 


2x-5- 

17              2 

2.    3  + 

1 
x  +  2 

5 

18 

2x-5  '  2x+l 

(^+2)2      (a 

:  +  2^3 

3.    6.^.+  l-l  +  4. 

X      x-'      x^ 

1 
x+1 

'■'^   '   x  +  1^: 

2 
(-r+l)2 

^.4i 

1         8       . 

(X-1)2 

5.    2x2-7-^  +  1- 

X        x3 

5 
x-1 

Art.  479 

;  page 

341. 

2. 

7            5x-3 
:c  _  1      x2  +  X  +  1 

3. 

.r2  +  2 

1 

x2  +  X  +  2 

3 

x  +  2 

2         x-10 
x-2      x2+4 

^■J 

,.^  +  6           3x-4 

+  X  +  1        x2  -  X  +  1 

^3- 

2          3x  +  4 
x-3      x2  +  3 

7.    - 

-l^h 

X                     X+1       . 
^X2+1         (x2+l)2 

Art.  481;   page  343. 
1.    x=y-y2  +  f-i/+  •••• 

2       a;  =  ^  y  J.  _  ,,2  _     ^      ,,3  _      ^"^      j,4    I     ... 

3^      27-^       243-^       2187^ 
3.    a:  =  2(/  +  6/  +  ^»/3  +  98/+  .••  • 


.    x=(^-l)-l(^-l)2  +  |(^-l)^-^(</-l)*+.... 


2 

5.  x  =  y+!/3  +  2^5+5yr+  .... 

-  o      ■  8    .,  .  28    3  ,  4G4    4  , 

6.  X  =  2«/  +  -//2+         ^3+  y4^. 


ANSWERS.  2H 

1  5    2  ,    29    3       19f)     4  , 

3^      27-^       243^       2187-^ 

^      3-^        15^       315^ 


Art.  484;   page  349. 
1   _3         3    -'    ,        7      -11    ,        77     -15    , 
^-    "5^      4  32  128  2048 

8.  l_l.+  1.2_il,3+4i^,_.... 

5        25  125         G25 

9.  a-5+5a-«x+15a-"a:2+35a-8a:3+70a-9ar*+     -  • 

10.  a:l-5xiy  +  5:r-i!/2+|r-|3/3  +  |:r-f3/*+  .... 

11.  a-r  _  2a-'^63  +  -aSfie  -  ^aV^g  +  ^^o^ii^ 

2  2  16 

12.  a-i  +  -a-5x-i  +  5a-9r-i+l^a-i3x-t  +  l^a-i-x-2+  ... 

2  8  16  128 

13.  x-^  -  4  X-*  y  +  16  x-62/2  -  64  x-^y^  +  256  x-i"?/* • 

14.  a:- "  +  Ix-^^ab  +  ^x"  ta262+  ^x"  ta^JS  +  ^x'a^i* 

2  2  8 

15.  aa  +  9a2y-3 +54at!/-3 +270a3y-i+  1215a2y"t4.  ...  . 

16.  1-4  .r(/-i  +  20  x2y-2  -  5?5  X33/-3  +  1^  x^y-* 

17.  32a5  +  20a3x-t+— ax-t+ Aa-ia:-2 ^q-s^:-!  +  .... 

4  32  1024 

18.  m2  +  37ntnf  +  ^-^miJ  +  5^mf  nV"  +  ^,„¥„5  +  ...  . 

2  2  8 

Art.  485;   page  350. 

2.    -^a-lxT.  3.   -364  mil.         4.    — a^.  5.    --^a"" "x«. 

2048  243  8192 

a     231    _,K,„  -  44      iA  _v  -         663    -jy.   _,, 

6.    a  2653.  7. x3  y    3.  8. x     2  w      • 

1024  19083        ^  8192  ^ 

9.    lLa-¥xW  10.   3003nV-V-.  n.    -^A^a-'^x-^. 

256  3. 

12.    715x-i*v-2i«-6.  13.  -^x-20. 

^  6561  16 

Art.  486 ;   page  351. 

2.    3.16228.       3.   10.04988.       4.    1.91294.       5.   3.02740.       6.   1.96799, 
7.    1.94729. 


24  COLLEGE   ALGEBRA. 

Art.  499  ;  page  355. 


2. 

1.6232. 

6. 

2.0491. 

10.   2.1673. 

14. 

3.7114. 

3. 

1.6532. 

7. 

2.1582. 

11.   2.5741. 

15. 

3.7814. 

4. 

1.7993. 

8. 

2.3343. 

12.   2.8363. 

16. 

4.0794. 

5. 

2.0212. 

9. 

2.1.303. 
Art.  501 ; 

13.   3.0545. 
page  356. 

17. 

4.2006. 

2. 

.5229. 

5. 

1.5441. 

8.   .2252. 

11. 

1.7602. 

3. 

.3589. 

6. 

.1182. 

9.   .7939. 

12. 

.8697. 

4. 

2.0458. 

7. 

2.3522. 
Art.  504; 

10.   2.3892. 
page  357. 

13. 

1.3380. 

3. 

.2863. 

8. 

.5880. 

13.    .3860. 

19. 

.6884. 

4. 

4.2255. 

9. 

3.2620. 

14.   .1909. 

20. 

.1840. 

5. 

.1398. 

10. 

.4225. 

16.   2.6145. 

21. 

.2215. 

6. 

4.5844. 

11. 

.0430. 

17.    .2601. 

22. 

.2494. 

7. 

.7194. 

12. 

.1165. 
Art.  506 ; 

18.   .1678. 
page  359. 

23. 

.1449. 

2. 

1.2922. 

5. 

7.4983-10. 

8.   6.6511-10. 

11. 

6.3588. 

3. 

0.6811. 

6. 

3.8663. 

9.   2.4804. 

12. 

.7964. 

4. 

9.5841  -  10, 

7. 

0.6074. 

10.   8.7905-10. 

13. 

.5063. 

Art.  512;  page  363. 

7.9.8878-10.      10.8.7164-10.  13.9.6055-10.  16.3.0155. 

8.  3.0237.              11.  1..3028.  14.  7.8560-10.  17.  8.9379-10. 

9.  0.5177.              12.  4.9659.  15.  0.7144.  18.  9.0010-10. 

Art.  513;  page  365. 

6.  1.646.                10    .003318.             13.  .2079.  16.  63329. 

7.  8886.                 11.  10221.                14.  44.48.  17.  .01301. 
9.   .01461.              12.  9.492.                 15.  .001109.  18.  502.9. 

Art.  518;   pages  368  to  370. 
1.   10.73.       2.  -2202.       3.  .2179.      4.  .01157.       5.  7.672.      6.  .6688. 
7.   -,3.908.        8.  3.500.        9.  -4.071.  10.  .2415.  11.  -.0725. 

12.    1.3587.        13.  -1.184.        14.  .000007038.        15.  4.642.       16.  .7567, 


ANSWERS. 

'IB 

ir. 

.006034. 

18.  1.442.          19.  3.162.          20.  .SCm.         21.  - 

-.4704. 

22. 

-  .3702. 

25.  5.883.     26.  .7024.     27.  1.502.     28.  .2510.     29. 

.9188 

30. 

-.7777. 

31.  .7295.          32.  .6357.         33.  -.6313.          34. 

.2979. 

35. 

98.50. 

36.  1.660.     37.  3.076.     38.  -11.34.     39.  .5881.     40. 
41.  .003229.            42.  .03345. 

1.805. 

Art.  519;   page  371. 
J.   .8115.      4.  .1853.       5.  -1.3852.       6.  -.2605. 


5  log( 


:.    -3.467. 

5.    11.193. 

6.    .9395. 

10.^. 

2 

n.  -1. 

-1 

log  a  —  2  log  6 

8.    ^-^&^ 9.3,-1.       10.-3.       11.  „  =  12&[^M£+1 

2  log  n  —  log  m  log  r 

12     „^IogrO--l)'S'+al-loga        jg    „^  logZ-loga  ^ 

logr  •  ■  log(S-a)-log(S-l) 

14.    „_log^-logrr/-(r-l);$']   I  ^^ 
logr 

Art.  520 ;   pages  371,  372. 

2.  3.7007.  3.   -.06552.        4. 
7.   -1.8204.  9.   5. 

Art.  529  ;   page  377. 

1.   4.605.  2.   -11.51.  3.   4.480.  4.    7.189.  5.   -1.068. 

6.  -2.458.  7.  5.  8.  4.  9.  6.  10.  20.  11.    7. 

Art.  538 ;   pages  383,  384. 

1.   $2853.75.       2.   $702.86.       3.   5i.      4.   ,^326.       5.   4.       6.    14.198. 

7.  16.01.       8.   $647.14.      9.   $2076.40.       10.   $2959.18.       11.   $5340. 

12.   $327.79.         13.   $4588.         14.   $277.         15.   $576.50. 

Art.  547 ;   pages  389  to  391. 

3.  8910720.         4.   40320.         5.   20389320.        6.   2002.        7.    18564. 

8.  170544.  9.  7893600.  10.  5040.  11.  60480.  12.  15890700. 
13.  840;  210;  5040;  13699.  14.  12-5970.  15.  220.  16.  27216. 
17.   21.      18.  56.     20.  15840.     21.  121030.     22.  10080.     23.  10584000. 

24.    15120.     25.  576.     26.  3303300.     27.  720.      28.  2700.      29.  JII_ 

my 


26  COLLEGE   ALGEBRA. 

Art.  548;   pag^e  392. 
1.   3780.       2.    1663200.       3.   G0060.       4.   360360.       5.   72.       6.   18. 

Art.  553;  pages  395,  396. 

1   ^^  \  2 .  ^i  ^  2   .  .  1   .  ,  X  24  .  s  50   .  .^  4   ,  s  100 

1.  (a.)  -  ;  (6.)  — -;  (c.)  —  ;  ((/.)  —  ;  (e.)  ;  (  f.) ;  [q.) 

V  J  ,_^,  K   ^  21  ^  ^91^  ^  91  ^  ^  1001  ^-^  ^  143  ^^  ^  1001 


'I- 

4.  ^1. 

30 

7    4 

5.  («-)t|7;  (^0 

115 

8.  11. 

24 

1331 
2300 

6.  ^'L. 

20825 

270725 

Art.  556; 

pages  398,  399. 

^•1 

3.  $2.50. 
8. 

4.  $50.    5.  — 
108 

85  cents. 

6. 

1 
8' 

7.  $3. 

Art.  564;   pages  405,  406. 


1. 

1 
32* 

2. 

4 
9" 

3   125.    4  23. 
3888       048 

5.  16. 

5525 

«•! 

7. 

63 

250' 

8. 

1 
60* 

9.  A.    10.  -81. 
14       136 

11.   47  . 

559872 

-  4i9- 

13. 

33 

1000 

14. 

2072      jg   5 
3125         18 

16.  11-2. 
243 

17.   20. 
91 

18. 

45927 
50000 

--i^^'i 

20.  A,|;B, 

9  .  C  11 

28 '   '  56' 

Art,  578;   pages  416,  417. 

^'   ^  +  2^lT3^IT^^^'^'^°"^^^^^°*'l|- 

_      1     1     1     1     1    1     r,,  ,15 

2. • ;  oth  convergent,  — 

1  +  3+1  +  3+1  +  3  ^      '19 

3.   3+— —  —  —  —— 1;  5th  convergent,  — . 

1+1+1+1+3+2+2  ^      '   5 

.       1     1     1     1     1     1     1    1     r,,  .8 

4. — ;  5th  convergent,  — 

1+2+1+2+1+2+1+2  ^         11 

-     „  ,     1     1     1     1     1     1    1     r,.  ,85 

5.    2  ^ • :  5th  convergent,  — 

3  +  2+1  +  3  +  2+1  +  2  ^         37 


ANSWERS.  27 

fl     1  .J_J_J_J_  J- -i.  J- J- 1;  5th  convergent,  |. 
'*•    ^■^1  +  1+1  +  1  +  1+1  +  1  +  1  +  2  5 

7  14.J_J__LJL-1-J-^;  5th  convergent,  — • 
^3+1  +  3+1  +  3+1  +  3'  19 

8  -i_  -L  J_  —  —  1 ;  5th  convergent,  -— • 

2  +  3  +  4  +  5  +  G+7  157 

1       1        .  u  ,    161        1         1 

9-   2  +  j-^  ^TTT:  ;  4th  convergent,  —  ,  ^^,  ^^• 

10-  3  +  ^  ^. ;  4th  convergent,  || ;  ^.  ^• 

11.,         ^  2177    1     1 

11-  4  +  ^  g-:^ ;  4th  convergent,  -^ ;  g^,  ^^^. 

12.  1  +  -^^ ^ ;  4th  convergent,  -  ;  — ,  — • 

1  +  2+  ••■  4        /i      lii 

11.,  ,   199        1         1 

13.  3  +  ^^^;  4th  convergent,  — ;  ^^,  ^q" 

14.  -1  +  ^  ^ :  ^'^  convergent,  ^  ;  ^.  ^• 

11.,  ,   161       1       1 

15-    4  +  i  ^:;:^ ;  4th  convergent,  -^  i  ^.  g^' 

16.    2  +  ^  ^  ^  j^ ;  4th  convergent,  | ;  1  ^. 

11111        .  ,  ,    13       1      1 

^^-    ^  +  ^  ^  rr  ^  3TT.  =  4^^^  convergent,  -  ;  -.  -• 

1111        .  ,  ,    15      1     1 

18.  3  +  —  -!-  -— ;  4th  convergent,  —  ,  —,  — • 

1  +  2+  1  +  6+  -  4      ^1    i^ 

19.  5  +  J_  J-  J 1 — ;  4th  convergent,  -j- ;  ^.  T^' 

^1  +  2+1  +  10+-  4      21   l.i 

1111  -,  ^71      11 

20     11  4-  _  -i-  -^ ^— ;  4th  convergent,  —  ;  77.  ^• 

1  +  4+1  +  22  +  -  6      55  30 


21 


1111  -  ,  X    71      1 

11  4.  J-  _i-  -i ^— ;  4th  convergent,  —  ;  —. 

1  +  4+1 +  22  +  -  6      55 

-3+ Vl5  03     3  + V5 

2  '  '2 
22.    -2  +  2V2.                          24.    _1  +  2V6. 

25.    3+-LJ--f 
7  +  15+1  +  •■ 


+  2V2.  24.    -H-2V 

111.,  ^355 

-1 1 —  ■  4th  convergent,  -— -• 

7  +  15+I  +  -  113 


28  COLLEGE    ALGEBRA. 


1111111         76 


2  +  3  +  3  +  3+1  +  1  +  7  + ■•■'  175'  262325'  231700 

11111111         193         1  1 


27.  2  + 


1  +  2+1  +  1  +  4+1  +  1  + 19 +  •■•'  71'  103589'  98548 


28.  3  +  —  ^— ;  5th  convergent,  — • 

1  +  5+-  ^  41 

29.  3  +  —  -^^ — - — —  -i— ;  4th  convergent,  — • 

1+1+1+1+6+ -  ^  3 

30.  -i-J- J-^^ L_;  4th  con%'ergent,  — • 

5+ 1  +  2+1 +10+-  17 

31.  3a  +  _i 1 —  ;  4th  convergent,  "^"^  +  24a2+l^ 

2a+6a  +  -  24a3  +  4a 

32.  4  + J — ^ — LJ_Jl_^-;  4t]i  convergent,  —• 

1  +  2  +  4+2+1  +  8+ -  13 


2 


Art.  586  ;  page  422. 

;    (3'  -  20  .r'-i.     2.    — ^  +  ^        ;   [1  +  (-  2)'-i]  x- 


l-5a:  +  6a:-^  l  +  a:-2x2 

l-5x 


1  —  a:  —  2  a;2 

2+lx       .  (1  +  4.-1) (_,).-!. 
1  +  5  -r  +  4  a-2 


^  2  +  5x 

+  5.r  +  . 

^-    1     l""!;,   .'   [5(3)^-1 -2(5)-i]x^-i. 
1  —  8x  +  15x^ 

5-2.r       .  ^3(4)r--i+2(-2)'-i]z'-i. 


1  _  2  .r  —  8  a:2  ■ 


1  4-  5  X  —  14  a:^ 


l-5r 


1  +  8r+  ISx^ 


-;  [5(-5)'-i-4(-3)'-i]x'-l. 


-1  +  2^-3^^  ;  19 xT,  9xB.       10.         i  +  (^^  +  i-\      •  I8xs63x8 
I_a:  +  x2  +  x3  l+2x  +  3x-^  +  4x3 

11     . 2^-a^--4x^     _2868x-,  7686 x«. 

I4.3x-x2-5x3 


A^^  SWERS. 


29 


Art.  593  ;   pages  426,  427. 
1.    7.  2.    341 ;  1386.  3.    7  ;  147.  4.    520 ;  2548. 

5.    Inin  +  l).        6.    10,416.  7.    -  8624  ;- 31,920.         8.25,333. 

9.    ln(n+l)(n  +  2).  10.    J  «(3«  -  1)  ;  ^  «2(«  +  1).  U.    US.".. 

6  Z  z 

12.    -n(2n^  +  3mn  +  Sm-2).  13.    8640 ;  20,988. 


14.    in(n+l)(2n+l). 
0 


16. 


15.    ^n%n+iy. 
4 

I  (n +  !)(«  + 2);  ^^n(n  +  l)(n  +  2)(n  +  3).  17.   847. 


I  ?«2  +  6  win  +  2  n2  -  6  m  -  3  n  +  1)  ■ 


Art.  595  ;   page  428. 
2.    1.3891.  3.   4.12364.  4.    .04386. 

6.    6.12208.  7.    1.12319. 


5.    2.04414. 


Art.  600;   pages  430,  431. 

1.  39.         .    2.    154.  3.    309.  4.    ah  +  be  +  ca-a^ -b^ -A 

5.    abc  +  2fg/i  -  ap  -  be,'- -  ch-\  6.    1  +  a2  +  6^  +  c2. 

Art.  618  ;  pages  442,  443. 

2.  10.         3.   0.         4.    -653.  5.    («  _  6)(6  _  c)(c  -  a).         6.    1. 

7.    0.         8.    6.         9.    -15.         10.    0. 

11.  abcd-ab-ac  —  ad  —  bc  —  bd-cd  +  2(a  +  b  +  c+d)—S. 

12.  a*  +  6*  +  c*  -  2  a262  -  2  i2c2  _  2  c2a2. 

13 .  a4  +  ji  +  c4  +  c/i  -  2  (a262  +  a2c2  +  ^2^2  ^  yi^a.  +  62(/-2  +  c2 J2)  +  8  aicd 

14.    0.         15.    8ryr.         16.    -972.  17.    (a/ -  67/1  +  cn)2. 

18.    Sy-r^rs.         19.    x*  +  (a  +  6  +  c  +  fO  ^^• 

Art.  625  ;  page  449. 
1.    r,  2  ;  y,  -  1  ;  2,  3.  2.    x,  3  ;  y,  -  4  ;  2,  0. 

3.    a-,  -4  ;  ?/,  2  ;  z,  -  5.  4.    ar,  2  ;  »/,  -  1  ;  ^,  3 ;  m,  -4. 


28,  -27 
- 15,      20 


4,  19,  -17 
-15,  8,  10 
-    2,  -24,      20 


30 


COLLEGE   ALGEBRA. 


6,     1, 

2 

62  +  c2, 

ab,          ca 

12,  23, 

-10 

8. 

ab, 

c2  +  a2,      be 

20,     8, 

5 

ca. 

be,      a2  +  62 

Art.  635  ;   page  453. 


2.    4, 


4.    -, 
4 


7.    1,    -I. 

2        3 


8.    4  a,  2  a. 


.5±2\/3 


).    m  +  3,  -  m  -  2. 


Art.  636  ;   page  454. 

2.  a;8-10a:2  +  31a:-30  =  0.  4.    x* -2x^ -5x^  +  6x  =  0. 

3.  ar*  -  55 a;2 -  210 X- 216=^0.       8.    Ga;*-37x3-4x2+57x+ 18  =  0. 

6.  9x4  +  6x8 -59x2- 20a; +100  =  0. 

7.  20x*  +  21  x3  -  400x2  -  21  x  + 20=0. 

8.  24x4  +  22x3-17x2_23x-6=0. 

9.    x4-14x2  +  l=0.         10.    16x4+16x3-112x2-148x  +  19  =  0. 
11.    64x4  +  32x2+1  =  0. 


4.    -2,-3. 


Art.  639  ;   page  456. 


7.    9,  -  ( 


Art.  651 ;  pages  465,  466. 

3.  x3  -  15x2 -63x  + 297  =  0.  8.    x3- 40x  +  650  =  0. 

4.  x*  -  30x3 +  250x- 3125=0.  9,    ar*  +  30x3  _  48x2  -  1458  =  0. 

5.  27x3 -30x  + 28  =  0.  10.    x*  -  35x2- 90x  + 270  =  0. 

6.  24x4+15x3  +  50x2-625  =  0.  11.    x3  +  14x2  +  57x -4  =  0. 

7.  x3 +6x2 +  2x- 64  =  0.  12.    x3  _  4x2 -115x- 82  =  0. 

13.  x3  +  2x2-3x-23  =  0. 

14.  x*  -  13 x3  +  61  x2  -  116x  -  12  =  0. 

15.  x*  +  24x3  +  219x2 +895x  + 1376  =  0. 

16.  x4-25x3  +  204x2-694x  +  844  =  0. 

Art.  656  ;   page  470. 
5.    Two  positive,  two  negative.       8.    Tiiree  positive,  two  negative. 
7.    Two  positive,  tliree  negative.     9.    One  positive,  two  imaginary. 


ANSWERS.  31 

10.  Four  imaginary.  12.   One  positive,  four  imaginary. 

11.  One  negative,  four  imaginary.     13.   One  negative,  four  imaginary. 

14.   One  positive,  one  negative,  four  imaginary. 

Art.  662  ;  page  477. 
21.    15(3x-l)*.        14.    a:(5:r3-6).        15.    12ar-l.        16.    3x2-19. 

17.  18x5-15x*-8a:3  +  15x2-6x-2. 

18.  4x3+18x2  +  6x-8. 

19.  2(x2-2)(x+l)(3x2  +  2x-2). 

20.  4  (4  X  +  5)2(5  X- 4)3(35  x+ 13). 

21.  4(x2  +  x-l)(x2-x  +  l)(2x3_x+ 1). 

Art.  663  ;   page  478. 


2. 

Second,  4.          3.    Third,  6.          4.    Fourth,  72.          5.    Fourth,  24. 

6.    Fifth,  240.             7.   Fifth,  600. 

Art.  666 ;    page  480. 

2. 

2,2,-7.         3.    -1,-1,6.        4.    1    1    -l       5.    1,1.-4,-4. 

6.   -3,-3,-3,2.         7.   1,1,-2,-2,2.         8.    6,6,-2,-4. 

Art.  670;  page  482. 

1. 

4;_(14.V7).      3.   1  +  ^10;  -11.      5.    1  +  ^TI ;  -  (1  +  v^). 

2. 

6;  -(1+^9).      4.   8;-(l+V3).      6.    1  +  \/5  ;  -  (1  +  ^7). 

Art.  672  ;   page  484. 

2.  One  each  between  0  and  1,  4  and  5,  0  and  —  1. 

3.  One  each  between  1  and  2,  2  and  3,-1  and  —2,-2  and  —3. 

4.  One  each  between  1  and  2,  0  and  —  1,-8  and  —  9. 

5.  One  each  between  0  and  1,  1  and  2,  4  and  6,-1  and  —  2. 

Art.  673  ;   page  486. 

2.  One  each  between  1  and  2,  4  and  5,  0  and  —  1. 

3.  One  each  between  0  and  1,  0  and  —  1,  —  2  and  —  3. 

4.  One  each  between  1  and  2,  3  and  4,  0  and  -1,-3  and  —  4, 

5.  One  each  between  2  and  3,-1  and  —  2,  —3  and  -4,-4  and  —5. 


32  COLLEGE  ALGEBRA. 

Art.  680;  page  493. 

3.  One  each  between  1  and  2,  4  and  5,  and  —  1  and  —  2. 

4.  One  between  0  and  —  1 ;  two  imaginary  roots. 

5.  One  each  between  2  and  3,  0  and  —  1,  and  —  i  and  —  5. 
6-    Two  between  3  and  4  ;  one  between  ^  3  and  —  4. 

7.  Two  between  0  and  1 ;  one  each  between  2  and  3,  and  —  3  and  —  4. 

8.  One  each  between  0  and  1,  and  —  1  and  —  2 ;  two  imaginary  roots. 

9.  One  each  between  0  and  1,  and  1  and  2  ;  two  between  —  2  and  —  3. 
10.  One  each  between  —  2  and  —  3,  and  —  3  and  —  4  ;  two  imaginary 

roots. 

Art.  688 ;  page  497. 

7.  One  each  between  1  and  2,-1  and  —  2,  and  —  3  and  —  4. 

8.  One  each  between  2  and  3,  4  and  5,  and  0  and  —  1. 

9.  One  each  between  —  1  and  —2,-2  and  —  3,  and  —  5  and  —  6. 
10.   One  each  between  0  and  1,  1  and  2,  0  and  —  1,  and  —  1  and  —  2, 


Art.  692  ;   page  503. 

2  "       '  2' 


1.    1,3,4.        2.   -1,6,-5.         3.    -2,  ~^-  ^^'^.        4.    1,  -, -4. 


5.   2,8,-3.     6.    1, --,  -1.     7.    1,2,-2,-3.     8.   -1,2,-2, 
3 


1   3 

2'  2' 


2,-1.  10.   -1,-2,-3,-5.  11.   1,2,3, 


12.    --,'-,  2, -3.       13.    -1,3,4,5.      14.    1.     15.   4  _j^-l±Vl3 


2   2  3  6 


16.   -  1,  3, 


o  ±" 


Art.  700  ;  page  508. 


-1,-,?-  3.    1,-3  ±2  V2.  4.    ±1,-5, -1. 

'23  6 


6     1    g  -  1  ±  ^a-  —  2  g  -  3 


7.   1, -3±2V2, 


2  3       5       3 

:\/2l  o         1    -7±3\/5  l±^/^ 


2 


9.    1.2.1  -?,-!  10.   -1.-3.       1   -l*V-3 


2       4       3  '        '       3 

11     .  1  4  1  3  ^^5 
'    '  4'      2 


ANSWERS. 
Art.  702  ;    page  509. 


33 


J   -l+\/5±V-10-2V5     -l-V5±V-10  +  2\/5 
4  '  4  * 


3.   2, 


l+\/5i:V-10  +  2V5     1-V5±V-10-2V5 
4  '  4  ' 

1+V5±V-10-2V5      -1-V5+V-10+2V6 


2.  -4, 

5.  10,- 

8.  -2, 

11.  4,= 


Art.  709;  page  512. 
2±3a/^.  3.   G, -3, -3.  4.   2, -1±6\/^. 

-2±VI^.  6.   -1,  5±4V^.         7.   -7,3,3. 


-5+2V-3. 
5±3\/^ 


12. 


a/( 


-1,-1. 

O+VGQ' 


10.   -1, 


18 


€- 


3±V-3. 
2 
V69\ 


18    ; 


Art.  710;   page  513. 

1.    ^2  -  ^,  ^^-^^  ±  ^/i+y/2  ^/^^ 
2  2 


Art.  715  ;  page  518. 
2.    _7,  5,  _i,  3.  3.    1, -7,  3±2V'5.  4.   8, -4, -2±4\/2 

5.   -4,  2,  l+V^.        6.   -5, -3,  4±V3.        7.   4,-1,— 


31 


Art.  724 ;  page  525. 

1.   1.2016;  -1.33006.  2.   5.1346.  3.   -2.1768.  4.   .0946„ 

5.   -5.7683.        6.   3.2361.        7.   -2.1575.        8.   .6458. 

9.   2.7663  ;  -  .4663.         10.   2.2469  ;  .5549  ;  -  .8019. 

11.   1.8794;  -.3473;  -1.5321.        12.   -2.2134^.       13.   2.0473;  .5937. 

14.   1.3086;  -1,1366.        15.   2.3569;  2.69202;  -2.0489. 

16.   4.8977;  -3.6331  ;  -.7124;  -5522.         17.1.2599.        18.2.5713. 

19.    1.4953.  20.   2.1538. 


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